Theorem feq23i 6507

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 6497	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)

Syntax hints:   ↔ wb 208   = wceq 1533  ⟶wf 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951  df-fn 6357  df-f 6358

This theorem is referenced by:  ftpg  6917  hashf  13697  funcoppc  17144  cnextfval  22669  uhgr0  26857  lfgredgge2  26908  mbfmvolf  31524  eulerpartlemt  31629  ismgmOLD  35127  elghomOLD  35164  tendoset  37894  pwssplit4  39687  isomushgr  43990  lincdifsn  44478