Theorem nemtbir 3082

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 2989	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 326	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ≠ wne 2987

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 210  df-ne 2988

This theorem is referenced by:  opthwiener  5369  opthprc  5580  snnen2o  8691  cfpwsdom  9995  fprodn0f  15337  m1exp1  15717  pmtrsn  18639  gzrngunitlem  20156  logbmpt  25374  ex-id  28219  ex-mod  28234  sltval2  33276  sltsolem1  33293  nolt02o  33312  coss0  35879  ensucne0  40237  clsk1indlem4  40747  clsk1indlem1  40748  etransc  42925