Theorem necon1bd 2584

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon1bd.1	⊢ (φ → (A ≠ B → ψ))

Assertion

Ref	Expression
necon1bd	⊢ (φ → (¬ ψ → A = B))

Proof of Theorem necon1bd

Step	Hyp	Ref	Expression
1		necon1bd.1	. . 3 ⊢ (φ → (A ≠ B → ψ))
2	1	con3d 125	. 2 ⊢ (φ → (¬ ψ → ¬ A ≠ B))
3		nne 2520	. 2 ⊢ (¬ A ≠ B ↔ A = B)
4	2, 3	syl6ib 217	1 ⊢ (φ → (¬ ψ → A = B))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-ne 2518

This theorem is referenced by:  fvclss  5462