Theorem nemtbir 3036

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 2940	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 323	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ≠ wne 2938

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 207  df-ne 2939

This theorem is referenced by:  opthwiener  5524  opthprc  5753  ord2eln012  8534  snnen2oOLD  9262  0sdom1dom  9272  cfpwsdom  10622  fprodn0f  16024  m1exp1  16410  pmtrsn  19552  gzrngunitlem  21468  logbmpt  26846  sltval2  27716  sltsolem1  27735  nolt02o  27755  ex-id  30463  ex-mod  30478  coss0  38461  ensucne0  43519  clsk1indlem4  44034  clsk1indlem1  44035  etransc  46239