Theorem nemtbir 3061

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 2967	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 326	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1568   ≠ wne 2965

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 2966

This theorem is referenced by:  opthwiener  5501  opthprc  5729  ord2eln012  8485  0sdom1dom  9209  cfpwsdom  10572  fprodn0f  16048  m1exp1  16437  pmtrsn  19592  gzrngunitlem  21565  logbmpt  26933  ltsval2  27800  ltssolem1  27819  nolt02o  27839  ex-id  30755  ex-mod  30770  coss0  39168  ensucne0  44207  clsk1indlem4  44722  clsk1indlem1  44723  etransc  46949