Theorem necon2bi 2419

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 2385	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
3	2	con2i 628	1 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364   ≠ wne 2364

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

This theorem depends on definitions:  df-bi 117  df-ne 2365

This theorem is referenced by:  minel  3508  rzal  3544  difsnb  3761  fin0  6941  0npi  7373  0nsr  7809  renfdisj  8079  nltpnft  9880  ngtmnft  9883  xrrebnd  9885  hashnncl  10866  rennim  11146  pceq0  12460