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 326	1 ⊢ ¬ 𝜑

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ≠ wne 2959

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 2960

This theorem is used by:  opthwiener  5491  opthprc  5719  ord2eln012  8530  0sdom1dom  9262  cfpwsdom  10693  fprodn0f  16184  m1exp1  16572  pmtrsn  19760  gzrngunitlem  21765  logbmpt  27143  ltsval2  28040  ltssolem1  28059  nolt02o  28079  ex-id  31062  ex-mod  31077  coss0  39692  ensucne0  44720  clsk1indlem4  45234  clsk1indlem1  45235  etransc  47496