Theorem nemtbir 3044

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 2948	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 323	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ≠ wne 2946

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 207  df-ne 2947

This theorem is referenced by:  opthwiener  5533  opthprc  5764  ord2eln012  8553  snnen2oOLD  9290  0sdom1dom  9301  cfpwsdom  10653  fprodn0f  16039  m1exp1  16424  pmtrsn  19561  gzrngunitlem  21473  logbmpt  26849  sltval2  27719  sltsolem1  27738  nolt02o  27758  ex-id  30466  ex-mod  30481  coss0  38435  ensucne0  43491  clsk1indlem4  44006  clsk1indlem1  44007  etransc  46204