Theorem feq23 6667

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 6665	. 2 ⊢ (𝐴 = 𝐶 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐵))
2		feq3 6666	. 2 ⊢ (𝐵 = 𝐷 → (𝐹:𝐶⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
3	1, 2	sylan9bb 517	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ⟶wf 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-cleq 2753  df-ss 3919  df-fn 6519  df-f 6520

This theorem is referenced by:  feq23i  6680  ismgmOLD  38310  ismndo2  38334  rngomndo  38395  seff  44846