Theorem nemtbir 2465

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		df-ne 2377	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbi 145	. 2 ⊢ ¬ 𝐴 = 𝐵
4		nemtbir.2	. 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
5	3, 4	mtbir 673	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   ↔ wb 105   = wceq 1373   ≠ wne 2376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2377

This theorem is referenced by:  opthprc  4726  php5  6955  snnen2oprc  6957  djulclb  7157  ismkvnex  7257  sucpw1nel3  7345  m1exp1  12212  pwle2  15935