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 1569   ≠ 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  5486  opthprc  5714  ord2eln012  8505  0sdom1dom  9237  cfpwsdom  10669  fprodn0f  16158  m1exp1  16546  pmtrsn  19733  gzrngunitlem  21738  logbmpt  27116  ltsval2  28013  ltssolem1  28032  nolt02o  28052  ex-id  31035  ex-mod  31050  coss0  39502  ensucne0  44530  clsk1indlem4  45044  clsk1indlem1  45045  etransc  47293