Theorem necon1i 2973

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

Hypothesis

Ref	Expression
necon1i.1	⊢ (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)

Assertion

Ref	Expression
necon1i	⊢ (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)

Proof of Theorem necon1i

Step	Hyp	Ref	Expression
1		df-ne 2940	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1i.1	. . 3 ⊢ (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)
3	1, 2	sylbir 235	. 2 ⊢ (¬ 𝐴 = 𝐵 → 𝐶 = 𝐷)
4	3	necon1ai 2967	1 ⊢ (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ≠ wne 2939

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 207  df-ne 2940

This theorem is referenced by: (None)