Theorem nemtbir 3037

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 2941	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 323	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1539   ≠ wne 2939

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 2940

This theorem is referenced by:  opthwiener  5518  opthprc  5748  ord2eln012  8536  snnen2oOLD  9265  0sdom1dom  9275  cfpwsdom  10625  fprodn0f  16028  m1exp1  16414  pmtrsn  19538  gzrngunitlem  21451  logbmpt  26832  sltval2  27702  sltsolem1  27721  nolt02o  27741  ex-id  30454  ex-mod  30469  coss0  38481  ensucne0  43547  clsk1indlem4  44062  clsk1indlem1  44063  etransc  46303