Theorem feq12i 6577

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 2738	. 2 ⊢ 𝐶 = 𝐶
4		feq123 6574	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
5	1, 2, 3, 4	mp3an 1459	1 ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)

Syntax hints:   ↔ wb 205   = wceq 1539  ⟶wf 6414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-fun 6420  df-fn 6421  df-f 6422

This theorem is referenced by:  climlimsupcex  43200