Theorem necon4bbiddc 2410

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

Hypothesis

Ref	Expression
necon4bbiddc.1	⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))))

Assertion

Ref	Expression
necon4bbiddc	⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 = 𝐵))))

Proof of Theorem necon4bbiddc

Step	Hyp	Ref	Expression
1		necon4bbiddc.1	. . . . . 6 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))))
2		bicom 139	. . . . . 6 ⊢ ((¬ 𝜓 ↔ 𝐴 ≠ 𝐵) ↔ (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))
3	1, 2	syl8ib 165	. . . . 5 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))))
4	3	com23 78	. . . 4 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))))
5	4	necon4abiddc 2409	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ 𝜓))))
6	5	com23 78	. 2 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ 𝜓))))
7		bicom 139	. 2 ⊢ ((𝐴 = 𝐵 ↔ 𝜓) ↔ (𝜓 ↔ 𝐴 = 𝐵))
8	6, 7	syl8ib 165	1 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 = 𝐵))))

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: (None)