Theorem necon3ai 2416

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 2368	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3ai.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	con3i 633	. 2 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝜑)
4	1, 3	sylbi 121	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364   ≠ wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2368

This theorem is referenced by:  nelsn  3657  disjsn2  3685  0nelxp  4691  fvunsng  5756  map0b  6746  difinfsnlem  7165  hashprg  10900  gcd1  12154  gcdzeq  12189  phimullem  12393  pcgcd1  12497  pc2dvds  12499  pockthlem  12525  znrrg  14216  mpodvdsmulf1o  15226  2sqlem8  15364