Theorem necon1abiddc 2409

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

Hypothesis

Ref	Expression
necon1abiddc.1	⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))

Assertion

Ref	Expression
necon1abiddc	⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓)))

Proof of Theorem necon1abiddc

Step	Hyp	Ref	Expression
1		necon1abiddc.1	. . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))
2	1	con1biddc 876	. 2 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝐴 = 𝐵 ↔ 𝜓)))
3		df-ne 2348	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	3	bibi1i 228	. 2 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝜓) ↔ (¬ 𝐴 = 𝐵 ↔ 𝜓))
5	2, 4	syl6ibr 162	1 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 834   = wceq 1353   ≠ wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-ne 2348

This theorem is referenced by:  necon2abiddc  2413