Theorem necon2i 2828

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 2799	. 2 ⊢ (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)
3	2	necon2ai 2823	1 ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   = wceq 1483   ≠ wne 2794

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 197  df-ne 2795

This theorem is referenced by:  cmpfi  21211  mcubic  24574  cubic2  24575  2sqlem11  25154  ovoliunnfl  33451  voliunnfl  33453  volsupnfl  33454  mncn0  37709  aaitgo  37732