Theorem nemtbir 2397

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 2309	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbi 144	. 2 ⊢ ¬ 𝐴 = 𝐵
4		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
5	3, 4	mtbir 660	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 104   = wceq 1331   ≠ wne 2308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2309

This theorem is referenced by:  opthprc  4590  php5  6752  snnen2oprc  6754  djulclb  6940  ismkvnex  7029  m1exp1  11598  pwle2  13193