Theorem necon2bbiddc 2373

Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon2bbiddc.1	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 ≠ 𝐵)))

Assertion

Ref	Expression
necon2bbiddc	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))

Proof of Theorem necon2bbiddc

Step	Hyp	Ref	Expression
1		necon2bbiddc.1	. . . 4 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 ≠ 𝐵)))
2		bicom 139	. . . 4 ⊢ ((𝜓 ↔ 𝐴 ≠ 𝐵) ↔ (𝐴 ≠ 𝐵 ↔ 𝜓))
3	1, 2	syl6ib 160	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜓)))
4	3	necon1bbiddc 2369	. 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))
5		bicom 139	. 2 ⊢ ((¬ 𝜓 ↔ 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ↔ ¬ 𝜓))
6	4, 5	syl6ib 160	1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 819   = wceq 1331   ≠ wne 2306

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-ne 2307

This theorem is referenced by: (None)