Theorem necon1ai 2559

Description: Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.)

Hypothesis

Ref	Expression
necon1ai.1	⊢ (¬ φ → A = B)

Assertion

Ref	Expression
necon1ai	⊢ (A ≠ B → φ)

Proof of Theorem necon1ai

Step	Hyp	Ref	Expression
1		df-ne 2519	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon1ai.1	. . 3 ⊢ (¬ φ → A = B)
3	2	con1i 121	. 2 ⊢ (¬ A = B → φ)
4	1, 3	sylbi 187	1 ⊢ (A ≠ B → φ)

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:  necon1i  2561  tz6.12i  5349  elfvdm  5352  elovex12  5649