Theorem necon4biddc 2411

Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)

Hypothesis

Ref	Expression
necon4biddc.1	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))

Assertion

Ref	Expression
necon4biddc	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))))

Proof of Theorem necon4biddc

Step	Hyp	Ref	Expression
1		necon4biddc.1	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))
2		df-ne 2337	. . . 4 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	2	bibi2i 226	. . 3 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐷))
4	1, 3	syl8ib 165	. 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐷))))
5	4	necon4abiddc 2409	1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 824   = 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  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-ne 2337

This theorem is referenced by:  nebidc  2416