Theorem nemtbir 3055

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 2961	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 325	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1562   ≠ wne 2959

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 2960

This theorem is referenced by:  opthwiener  5485  opthprc  5713  ord2eln012  8468  0sdom1dom  9192  cfpwsdom  10544  fprodn0f  16023  m1exp1  16412  pmtrsn  19561  gzrngunitlem  21486  logbmpt  26855  ltsval2  27722  ltssolem1  27741  nolt02o  27761  ex-id  30638  ex-mod  30653  coss0  39073  ensucne0  44110  clsk1indlem4  44625  clsk1indlem1  44626  etransc  46862