Theorem necon2bi 2455

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

Hypothesis

Ref	Expression
necon2bi.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Assertion

Ref	Expression
necon2bi	⊢ (𝐴 = 𝐵 → ¬ 𝜑)

Proof of Theorem necon2bi

Step	Hyp	Ref	Expression
1		necon2bi.1	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2	1	neneqd 2421	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
3	2	con2i 630	1 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395   ≠ wne 2400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-ne 2401

This theorem is referenced by:  minel  3553  rzal  3589  difsnb  3811  fin0  7060  0npi  7516  0nsr  7952  renfdisj  8222  nltpnft  10027  ngtmnft  10030  xrrebnd  10032  hashnncl  11034  rennim  11534  pceq0  12866