Theorem feq23i 5275

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 5266	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
4	1, 2, 3	mp2an 423	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)

Syntax hints:   ↔ wb 104   = wceq 1332  ⟶wf 5127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089  df-fn 5134  df-f 5135

This theorem is referenced by:  ftpg  5612