Theorem nemtbir 3038

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 2942	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 322	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1541   ≠ wne 2940

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 206  df-ne 2941

This theorem is referenced by:  opthwiener  5514  opthprc  5740  ord2eln012  8499  snnen2oOLD  9229  0sdom1dom  9240  cfpwsdom  10581  fprodn0f  15939  m1exp1  16323  pmtrsn  19428  gzrngunitlem  21210  logbmpt  26517  sltval2  27383  sltsolem1  27402  nolt02o  27422  ex-id  29942  ex-mod  29957  coss0  37652  ensucne0  42582  clsk1indlem4  43097  clsk1indlem1  43098  etransc  45298