Theorem necon2bi 2563

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

Hypothesis

Ref	Expression
necon2bi.1	⊢ (φ → A ≠ B)

Assertion

Ref	Expression
necon2bi	⊢ (A = B → ¬ φ)

Proof of Theorem necon2bi

Step	Hyp	Ref	Expression
1		necon2bi.1	. . 3 ⊢ (φ → A ≠ B)
2	1	neneqd 2533	. 2 ⊢ (φ → ¬ A = B)
3	2	con2i 112	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:  necon4i  2577  minel  3607  rzal  3652  difsnb  3851  nulnnc  6119