Theorem nemtbir 3039

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 2943	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 323	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1542   ≠ wne 2941

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 206  df-ne 2942

This theorem is referenced by:  opthwiener  5515  opthprc  5741  ord2eln012  8497  snnen2oOLD  9227  0sdom1dom  9238  cfpwsdom  10579  fprodn0f  15935  m1exp1  16319  pmtrsn  19387  gzrngunitlem  21010  logbmpt  26293  sltval2  27159  sltsolem1  27178  nolt02o  27198  ex-id  29718  ex-mod  29733  coss0  37397  ensucne0  42328  clsk1indlem4  42843  clsk1indlem1  42844  etransc  45047