Theorem nemtbir 3051

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 2957	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 326	1 ⊢ ¬ 𝜑

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ≠ wne 2955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ne 2956

This theorem is used by:  opthwiener  5483  opthprc  5711  ord2eln012  8483  0sdom1dom  9215  cfpwsdom  10640  fprodn0f  16125  m1exp1  16513  pmtrsn  19694  gzrngunitlem  21699  logbmpt  27079  ltsval2  27946  ltssolem1  27965  nolt02o  27985  ex-id  30968  ex-mod  30983  coss0  39421  ensucne0  44473  clsk1indlem4  44988  clsk1indlem1  44989  etransc  47215