Theorem feq23i 6697

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 6684	. 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 209   = wceq 1567  ⟶wf 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-ss 3930  df-fn 6537  df-f 6538

This theorem is referenced by:  ftpg  7151  hashf  14370  funcoppc  17928  cnextfval  24184  uhgr0  29360  lfgredgge2  29411  mbfmvolf  34597  eulerpartlemt  34702  ismgmOLD  38384  elghomOLD  38421  tendoset  41418  pwssplit4  43703  gricushgr  48566  uspgrlimlem2  48638  lincdifsn  49084