Theorem necon1bi 2560

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon1bi.1	⊢ (A ≠ B → φ)

Assertion

Ref	Expression
necon1bi	⊢ (¬ φ → A = B)

Proof of Theorem necon1bi

Step	Hyp	Ref	Expression
1		necon1bi.1	. . 3 ⊢ (A ≠ B → φ)
2	1	con3i 127	. 2 ⊢ (¬ φ → ¬ A ≠ B)
3		nne 2521	. 2 ⊢ (¬ A ≠ B ↔ A = B)
4	2, 3	sylib 188	1 ⊢ (¬ φ → 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:  mapprc  6005