Theorem necon3ai 2451

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 2403	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3ai.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	con3i 637	. 2 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝜑)
4	1, 3	sylbi 121	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1397   ≠ wne 2402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2403

This theorem is referenced by:  nelsn  3704  disjsn2  3732  0nelxp  4753  fvunsng  5848  map0b  6856  difinfsnlem  7298  hashprg  11073  gcd1  12560  gcdzeq  12595  phimullem  12799  pcgcd1  12903  pc2dvds  12905  pockthlem  12931  znrrg  14677  mpodvdsmulf1o  15717  2sqlem8  15855