Theorem necon2bi 2390

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 2356	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
3	2	con2i 617	1 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1343   ≠ wne 2335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2336

This theorem is referenced by:  minel  3469  rzal  3505  difsnb  3715  fin0  6847  0npi  7250  0nsr  7686  renfdisj  7954  nltpnft  9746  ngtmnft  9749  xrrebnd  9751  hashnncl  10705  rennim  10940  pceq0  12249