Theorem necon2bd 2364

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Hypothesis

Ref	Expression
necon2bd.1	⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))

Assertion

Ref	Expression
necon2bd	⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))

Proof of Theorem necon2bd

Step	Hyp	Ref	Expression
1		necon2bd.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))
2		df-ne 2307	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	syl6ib 160	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵))
4	3	con2d 613	1 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ≠ wne 2306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2307

This theorem is referenced by:  disjiun  3919  map0g  6575  nneo  9147  zeo2  9150  bezoutr1  11710  coprm  11811  sqrt2irr  11829  dfphi2  11885