Theorem nemtbir 3094

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 3001	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 315	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1656   ≠ wne 2999

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 199  df-ne 3000

This theorem is referenced by:  opthwiener  5202  opthprc  5404  snnen2o  8424  cfpwsdom  9728  m1exp1  15474  pmtrsn  18297  gzrngunitlem  20178  logbmpt  24935  ex-id  27845  ex-mod  27860  sltval2  32343  sltsolem1  32360  nolt02o  32379  coss0  34772  clsk1indlem4  39177  clsk1indlem1  39178  etransc  41288