Theorem necon2ai 2434

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 630	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
3		df-ne 2381	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	sylibr 134	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1375   ≠ wne 2380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-ne 2381

This theorem is referenced by:  necon2i  2436  neneqad  2459  intexr  4213  iin0r  4232  tfrlemisucaccv  6441  pm54.43  7331  renepnf  8162  renemnf  8163  lt0ne0d  8628  nnne0  9106  nn0nepnf  9408  hashennn  10969  bj-intexr  16181