Theorem necon2abiddc 2311

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

Hypothesis

Ref	Expression
necon2abiddc.1	⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))

Assertion

Ref	Expression
necon2abiddc	⊢ (𝜑 → (DECID 𝜓 → (𝜓 ↔ 𝐴 ≠ 𝐵)))

Proof of Theorem necon2abiddc

Step	Hyp	Ref	Expression
1		necon2abiddc.1	. . . 4 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
2		bicom 138	. . . 4 ⊢ ((𝐴 = 𝐵 ↔ ¬ 𝜓) ↔ (¬ 𝜓 ↔ 𝐴 = 𝐵))
3	1, 2	syl6ib 159	. . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))
4	3	necon1abiddc 2307	. 2 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓)))
5		bicom 138	. 2 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝜓) ↔ (𝜓 ↔ 𝐴 ≠ 𝐵))
6	4, 5	syl6ib 159	1 ⊢ (𝜑 → (DECID 𝜓 → (𝜓 ↔ 𝐴 ≠ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  DECID wdc 775   = wceq 1284   ≠ wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776  df-ne 2246

This theorem is referenced by: (None)