Theorem necon2i 2995

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Hypothesis

Ref	Expression
necon2i.1	⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)

Assertion

Ref	Expression
necon2i	⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)

Proof of Theorem necon2i

Step	Hyp	Ref	Expression
1		necon2i.1	. . 3 ⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)
2	1	neneqd 2966	. 2 ⊢ (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)
3	2	necon2ai 2990	1 ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2961

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ne 2962

This theorem is used by:  cmpfi  23602  mcubic  27049  cubic2  27050  2sqlem11  27630  zarcmplem  34302  ovoliunnfl  38354  voliunnfl  38356  volsupnfl  38357  mncn0  43907  aaitgo  43930  usgrexmpl2trifr  48843