Theorem necon2i 2992

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 2963	. 2 ⊢ (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)
3	2	necon2ai 2987	1 ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   = wceq 1570   ≠ wne 2958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-ne 2959

This theorem is referenced by:  cmpfi  23546  mcubic  26993  cubic2  26994  2sqlem11  27574  zarcmplem  34252  ovoliunnfl  38294  voliunnfl  38296  volsupnfl  38297  mncn0  43849  aaitgo  43872  usgrexmpl2trifr  48785