Theorem nemtbir 3032

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 2938	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 325	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1548   ≠ wne 2936

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 2937

This theorem is referenced by:  opthwiener  5458  opthprc  5685  ord2eln012  8426  0sdom1dom  9150  cfpwsdom  10502  fprodn0f  15951  m1exp1  16340  pmtrsn  19489  gzrngunitlem  21411  logbmpt  26774  ltsval2  27642  ltssolem1  27661  nolt02o  27681  ex-id  30526  ex-mod  30541  coss0  38951  ensucne0  43988  clsk1indlem4  44503  clsk1indlem1  44504  etransc  46740