Theorem nemtbir 3028

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 2932	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 322	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1534   ≠ wne 2930

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 2931

This theorem is referenced by:  opthwiener  5520  opthprc  5746  ord2eln012  8527  snnen2oOLD  9261  0sdom1dom  9272  cfpwsdom  10627  fprodn0f  15993  m1exp1  16378  pmtrsn  19517  gzrngunitlem  21429  logbmpt  26816  sltval2  27686  sltsolem1  27705  nolt02o  27725  ex-id  30367  ex-mod  30382  coss0  38177  ensucne0  43196  clsk1indlem4  43711  clsk1indlem1  43712  etransc  45904