Theorem necon1abiddc 2371

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 862	. 2 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝐴 = 𝐵 ↔ 𝜓)))
3		df-ne 2310	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	3	bibi1i 227	. 2 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝜓) ↔ (¬ 𝐴 = 𝐵 ↔ 𝜓))
5	2, 4	syl6ibr 161	1 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 820   = wceq 1332   ≠ wne 2309

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 817  df-dc 821  df-ne 2310

This theorem is referenced by:  necon2abiddc  2375