Theorem nemtbir 3053

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 2959	. 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 2957

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 2958

This theorem is used by:  opthwiener  5496  opthprc  5724  ord2eln012  8480  0sdom1dom  9204  cfpwsdom  10575  fprodn0f  16052  m1exp1  16440  pmtrsn  19595  gzrngunitlem  21593  logbmpt  26964  ltsval2  27831  ltssolem1  27850  nolt02o  27870  ex-id  30796  ex-mod  30811  coss0  39246  ensucne0  44283  clsk1indlem4  44798  clsk1indlem1  44799  etransc  47025