Theorem nemtbir 3042

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 2947	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 323	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1542   ≠ wne 2945

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 2946

This theorem is referenced by:  opthwiener  5432  opthprc  5652  snnen2o  8980  cfpwsdom  10341  fprodn0f  15699  m1exp1  16083  pmtrsn  19125  gzrngunitlem  20661  logbmpt  25936  ex-id  28794  ex-mod  28809  sltval2  33855  sltsolem1  33874  nolt02o  33894  coss0  36593  ensucne0  41115  clsk1indlem4  41624  clsk1indlem1  41625  etransc  43795