Theorem necon3ai 2389

Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon3ai.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
necon3ai	⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)

Proof of Theorem necon3ai

Step	Hyp	Ref	Expression
1		df-ne 2341	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3ai.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	con3i 627	. 2 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝜑)
4	1, 3	sylbi 120	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1348   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by:  nelsn  3618  disjsn2  3646  0nelxp  4639  fvunsng  5690  map0b  6665  difinfsnlem  7076  hashprg  10743  gcd1  11942  gcdzeq  11977  phimullem  12179  pcgcd1  12281  pc2dvds  12283  pockthlem  12308  2sqlem8  13753