Theorem necon4ddc 2412

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)

Hypothesis

Ref	Expression
necon4ddc.1	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)))

Assertion

Ref	Expression
necon4ddc	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵)))

Proof of Theorem necon4ddc

Step	Hyp	Ref	Expression
1		necon4ddc.1	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)))
2		df-ne 2341	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3		df-ne 2341	. . . 4 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
4	2, 3	imbi12i 238	. . 3 ⊢ ((𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
5	1, 4	syl6ib 160	. 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)))
6		condc 848	. 2 ⊢ (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷) → (𝐶 = 𝐷 → 𝐴 = 𝐵)))
7	5, 6	sylcom 28	1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 829   = wceq 1348   ≠ wne 2340

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-ne 2341

This theorem is referenced by: (None)