Theorem necon3ai 2469

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 2421	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3ai.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	con3i 641	. 2 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝜑)
4	1, 3	sylbi 121	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1402   ≠ wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  nelsn  3744  disjsn2  3772  0nelxp  4802  fvunsng  5909  map0b  6968  difinfsnlem  7439  hashprg  11251  gcd1  12766  gcdzeq  12801  phimullem  13005  pcgcd1  13109  pc2dvds  13111  pockthlem  13137  znrrg  14997  mpodvdsmulf1o  16110  2sqlem8  16254