Theorem feq23i 6703

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 6690	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ⟶wf 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-ss 3923  df-fn 6543  df-f 6544

This theorem is used by:  ftpg  7159  hashf  14392  funcoppc  17954  cnextfval  24270  uhgr0  29478  lfgredgge2  29529  mbfmvolf  34721  eulerpartlemt  34826  ismgmOLD  38559  elghomOLD  38596  tendoset  41591  pwssplit4  43874  gricushgr  48740  uspgrlimlem2  48812  lincdifsn  49261