Theorem nemtbir 2605

Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)

Hypotheses

Ref	Expression
nemtbir.1	⊢ A ≠ B
nemtbir.2	⊢ (φ ↔ A = B)

Assertion

Ref	Expression
nemtbir	⊢ ¬ φ

Proof of Theorem nemtbir

Step	Hyp	Ref	Expression
1		nemtbir.1	. . 3 ⊢ A ≠ B
2		df-ne 2519	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
3	1, 2	mpbi 199	. 2 ⊢ ¬ A = B
4		nemtbir.2	. 2 ⊢ (φ ↔ A = B)
5	3, 4	mtbir 290	1 ⊢ ¬ φ

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  ncvsq  6257