Theorem necon3bd 2298

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon3bd.1	⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))

Assertion

Ref	Expression
necon3bd	⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))

Proof of Theorem necon3bd

Step	Hyp	Ref	Expression
1		necon3bd.1	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))
2	1	con3d 596	. 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝐴 = 𝐵))
3		df-ne 2256	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	syl6ibr 160	1 ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1289   ≠ wne 2255

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580

This theorem depends on definitions:  df-bi 115  df-ne 2256

This theorem is referenced by:  nelne1  2345  nelne2  2346  nssne1  3080  nssne2  3081  disjne  3333  difsn  3569  nbrne1  3854  nbrne2  3855  ac6sfi  6594  indpi  6880  zneo  8817