Theorem nemtbir 3040

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 2945	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 323	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539   ≠ wne 2943

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 2944

This theorem is referenced by:  opthwiener  5428  opthprc  5651  ord2eln012  8327  snnen2oOLD  9010  cfpwsdom  10340  fprodn0f  15701  m1exp1  16085  pmtrsn  19127  gzrngunitlem  20663  logbmpt  25938  ex-id  28798  ex-mod  28813  sltval2  33859  sltsolem1  33878  nolt02o  33898  coss0  36597  ensucne0  41136  clsk1indlem4  41654  clsk1indlem1  41655  etransc  43824