Theorem nemtbir 3029

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 2935	. 2 ⊢ ¬ 𝐴 = 𝐵
3		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
4	2, 3	mtbir 323	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1542   ≠ wne 2933

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 2934

This theorem is referenced by:  opthwiener  5472  opthprc  5698  ord2eln012  8436  0sdom1dom  9160  cfpwsdom  10509  fprodn0f  15928  m1exp1  16317  pmtrsn  19465  gzrngunitlem  21404  logbmpt  26771  ltsval2  27641  ltssolem1  27660  nolt02o  27680  ex-id  30527  ex-mod  30542  coss0  38849  ensucne0  43914  clsk1indlem4  44429  clsk1indlem1  44430  etransc  46670