Theorem necon4bid 3013

Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)

Hypothesis

Ref	Expression
necon4bid.1	⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))

Assertion

Ref	Expression
necon4bid	⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))

Proof of Theorem necon4bid

Step	Hyp	Ref	Expression
1		necon4bid.1	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
2	1	necon2bbid 3011	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 ↔ ¬ 𝐴 ≠ 𝐵))
3		nne 2972	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	syl6rbb 280	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   = wceq 1601   ≠ wne 2968

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 199  df-ne 2969

This theorem is referenced by:  nebi  3048  znnenlemOLD  15344  rpexp  15836  norm-i  28558  trlid0b  36327