Theorem nemtbir 2877

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	1	neii 2784	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 312	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ≠ wne 2780

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 196  df-ne 2782

This theorem is referenced by:  opthwiener  4901  opthprc  5089  snnen2o  8034  cfpwsdom  9285  m1exp1  14931  pmtrsn  17762  gzrngunitlem  19630  logbmpt  24326  ex-id  26683  ex-mod  26698  sltval2  31053  sltsolem1  31067  clsk1indlem4  37362  clsk1indlem1  37363  etransc  39176