Positive relative consistency of CH and GCH

There is a fixed finite $\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{G}\mathrm{C}\mathrm{H}$ proof of CH: instantiate GCH at the first infinite initial ordinal and expand the selected cardinal notation. Hence appending this proof and replacing uses of the CH axiom gives a primitive-recursive map from any $\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{C}\mathrm{H}$ refutation to a $\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{G}\mathrm{C}\mathrm{H}$ refutation, and PA verifies the map by its finite line-prefix check. Therefore $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{G}\mathrm{C}\mathrm{H})$ implies $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{C}\mathrm{H})$.

Combining step 1.1 with F1 gives both implications from $\mathrm{Con}(\mathrm{Z}\mathrm{F})$. Separately, the inclusion of every certified ZF axiom in ZFC gives an identity-on-lines primitive-recursive map from ZF refutations to ZFC refutations. Thus PA proves $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C})\to \mathrm{Con}(\mathrm{Z}\mathrm{F})$. Composing this with the two implications already proved gives both conclusions from $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C})$ as claimed.