Theorem feq23 6568

Description: Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
feq23	⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))

Proof of Theorem feq23

Step	Hyp	Ref	Expression
1		feq2 6566	. 2 ⊢ (𝐴 = 𝐶 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐵))
2		feq3 6567	. 2 ⊢ (𝐵 = 𝐷 → (𝐹:𝐶⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
3	1, 2	sylan9bb 509	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ⟶wf 6414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-fn 6421  df-f 6422

This theorem is referenced by:  feq23i  6578  ismgmOLD  35935  ismndo2  35959  rngomndo  36020  seff  41816