Theorem necon3i 2384

Description: Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)

Hypothesis

Ref	Expression
necon3i.1	⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)

Assertion

Ref	Expression
necon3i	⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)

Proof of Theorem necon3i

Step	Hyp	Ref	Expression
1		necon3i.1	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)
2		id 19	. . 3 ⊢ ((𝐴 = 𝐵 → 𝐶 = 𝐷) → (𝐴 = 𝐵 → 𝐶 = 𝐷))
3	2	necon3d 2380	. 2 ⊢ ((𝐴 = 𝐵 → 𝐶 = 𝐷) → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵))
4	1, 3	ax-mp 5	1 ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)

Syntax hints:   → 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:  expnprm  12283  lgsne0  13579