Theorem nemtbir 2889

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 2796	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 313	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1483   ≠ wne 2794

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 197  df-ne 2795

This theorem is referenced by:  opthwiener  4976  opthprc  5167  snnen2o  8149  cfpwsdom  9406  m1exp1  15093  pmtrsn  17939  gzrngunitlem  19811  logbmpt  24526  ex-id  27291  ex-mod  27306  sltval2  31809  sltsolem1  31826  nolt02o  31845  clsk1indlem4  38342  clsk1indlem1  38343  etransc  40500