Theorem feq12i 6688

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 2731	. 2 ⊢ 𝐶 = 𝐶
4		feq123 6685	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
5	1, 2, 3, 4	mp3an 1461	1 ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)

Syntax hints:   ↔ wb 205   = wceq 1541  ⟶wf 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3426  df-v 3468  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-sn 4614  df-pr 4616  df-op 4620  df-br 5133  df-opab 5195  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-fun 6525  df-fn 6526  df-f 6527

This theorem is referenced by:  climlimsupcex  44170