Theorem necon2bi 2300

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 2266	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
3	2	con2i 589	1 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1284   ≠ wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-ne 2246

This theorem is referenced by:  minel  3305  rzal  3338  difsnb  3528  fin0  6369  0npi  6503  0nsr  6926  renfdisj  7172  nltpnft  8884  ngtmnft  8885  xrrebnd  8886  rennim  9888