Theorem feq12i 6648

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 2739	. 2 ⊢ 𝐶 = 𝐶
4		feq123 6645	. 2 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
5	1, 2, 3, 4	mp3an 1469	1 ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)

Syntax hints:   ↔ wb 207   = wceq 1547  ⟶wf 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-fun 6487  df-fn 6488  df-f 6489

This theorem is referenced by:  climlimsupcex  46212