Theorem feq23d 5221

Description: Equality deduction for functions. (Contributed by set.mm contributors, 8-Jun-2013.)

Hypotheses

Ref	Expression
feq23d.1	⊢ (φ → A = C)
feq23d.2	⊢ (φ → B = D)

Assertion

Ref	Expression
feq23d	⊢ (φ → (F:A–→B ↔ F:C–→D))

Proof of Theorem feq23d

Step	Hyp	Ref	Expression
1		feq23d.1	. 2 ⊢ (φ → A = C)
2		feq23d.2	. 2 ⊢ (φ → B = D)
3		feq23 5214	. 2 ⊢ ((A = C ∧ B = D) → (F:A–→B ↔ F:C–→D))
4	1, 2, 3	syl2anc 642	1 ⊢ (φ → (F:A–→B ↔ F:C–→D))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  –→wf 4778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-fn 4791  df-f 4792

This theorem is referenced by: (None)