Theorem necon4i 2978

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon4i.1	⊢ (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)

Assertion

Ref	Expression
necon4i	⊢ (𝐶 = 𝐷 → 𝐴 = 𝐵)

Proof of Theorem necon4i

Step	Hyp	Ref	Expression
1		necon4i.1	. . 3 ⊢ (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)
2	1	neneqd 2947	. 2 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐶 = 𝐷)
3	2	necon4ai 2974	1 ⊢ (𝐶 = 𝐷 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ≠ wne 2942

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 206  df-ne 2943

This theorem is referenced by:  unixp0  6175  scott0  9575  nn0opthi  13912