Theorem feq12i 6638

Description: Equality inference for functions. (Contributed by AV, 7-Feb-2021.)

Hypotheses

Ref	Expression
feq12i.1	⊢ 𝐹 = 𝐺
feq12i.2	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
feq12i	⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)

Proof of Theorem feq12i

Step	Hyp	Ref	Expression
1		feq12i.1	. 2 ⊢ 𝐹 = 𝐺
2		feq12i.2	. 2 ⊢ 𝐴 = 𝐵
3		eqid 2736	. 2 ⊢ 𝐶 = 𝐶
4		feq123 6635	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
5	1, 2, 3, 4	mp3an 1460	1 ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)

Syntax hints:   ↔ wb 205   = wceq 1540  ⟶wf 6469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-br 5090  df-opab 5152  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-fun 6475  df-fn 6476  df-f 6477

This theorem is referenced by:  climlimsupcex  43635