Theorem necon1bbiddc 2369

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

Hypothesis

Ref	Expression
necon1bbiddc.1	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜓)))

Assertion

Ref	Expression
necon1bbiddc	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))

Proof of Theorem necon1bbiddc

Step	Hyp	Ref	Expression
1		necon1bbiddc.1	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜓)))
2		df-ne 2307	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	bibi1i 227	. . 3 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝜓) ↔ (¬ 𝐴 = 𝐵 ↔ 𝜓))
4	1, 3	syl6ib 160	. 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 ↔ 𝜓)))
5	4	con1biddc 861	1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))

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:  necon2bbiddc  2373