Theorem nemtbir 3114

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 3020	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 325	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1537   ≠ wne 3018

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 209  df-ne 3019

This theorem is referenced by:  opthwiener  5406  opthprc  5618  snnen2o  8709  cfpwsdom  10008  fprodn0f  15347  m1exp1  15729  pmtrsn  18649  gzrngunitlem  20612  logbmpt  25368  ex-id  28215  ex-mod  28230  sltval2  33165  sltsolem1  33182  nolt02o  33201  coss0  35721  ensucne0  39902  clsk1indlem4  40401  clsk1indlem1  40402  etransc  42575