Theorem feq23i 5938

Description: Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
feq23i.1	⊢ 𝐴 = 𝐶
feq23i.2	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
feq23i	⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)

Proof of Theorem feq23i

Step	Hyp	Ref	Expression
1		feq23i.1	. 2 ⊢ 𝐴 = 𝐶
2		feq23i.2	. 2 ⊢ 𝐵 = 𝐷
3		feq23 5928	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
4	1, 2, 3	mp2an 704	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)

Syntax hints:   ↔ wb 195   = wceq 1475  ⟶wf 5786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-fn 5793  df-f 5794

This theorem is referenced by:  ftpg  6306  funcoppc  16307  cnextfval  21624  uhgra0v  25633  wlkntrllem1  25883  mbfmvolf  29449  eulerpartlemt  29554  ismgmOLD  32613  elghomOLD  32650  tendoset  34859  pwssplit4  36471  uhgr0  40290  lfgredgge2  40341  lincdifsn  41999