Theorem necon2ad 2565

Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon2ad.1	⊢ (φ → (A = B → ¬ ψ))

Assertion

Ref	Expression
necon2ad	⊢ (φ → (ψ → A ≠ B))

Proof of Theorem necon2ad

Step	Hyp	Ref	Expression
1		nne 2521	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
2		necon2ad.1	. . 3 ⊢ (φ → (A = B → ¬ ψ))
3	1, 2	syl5bi 208	. 2 ⊢ (φ → (¬ A ≠ B → ¬ ψ))
4	3	con4d 97	1 ⊢ (φ → (ψ → A ≠ B))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2517

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 2519

This theorem is referenced by:  necon2d  2567  nulnnn  4557  enadjlem1  6060