Theorem necon2ai 2390

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon2ai.1	⊢ (𝐴 = 𝐵 → ¬ 𝜑)

Assertion

Ref	Expression
necon2ai	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Proof of Theorem necon2ai

Step	Hyp	Ref	Expression
1		necon2ai.1	. . 3 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)
2	1	con2i 617	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
3		df-ne 2337	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	sylibr 133	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1343   ≠ wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  necon2i  2392  neneqad  2415  intexr  4129  iin0r  4148  tfrlemisucaccv  6293  pm54.43  7146  renepnf  7946  renemnf  7947  lt0ne0d  8411  nnne0  8885  nn0nepnf  9185  hashennn  10693  bj-intexr  13790