Theorem fneq2d 5177

Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
fneq2d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
fneq2d	⊢ (φ → (F Fn A ↔ F Fn B))

Proof of Theorem fneq2d

Step	Hyp	Ref	Expression
1		fneq2d.1	. 2 ⊢ (φ → A = B)
2		fneq2 5175	. 2 ⊢ (A = B → (F Fn A ↔ F Fn B))
3	1, 2	syl 15	1 ⊢ (φ → (F Fn A ↔ F Fn B))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   Fn wfn 4777

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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-fn 4791

This theorem is referenced by:  fneq12d  5178