Theorem necon1abiidc 2368

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

Hypothesis

Ref	Expression
necon1abiidc.1	⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵))

Assertion

Ref	Expression
necon1abiidc	⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑))

Proof of Theorem necon1abiidc

Step	Hyp	Ref	Expression
1		df-ne 2309	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1abiidc.1	. . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵))
3	2	con1biidc 862	. 2 ⊢ (DECID 𝜑 → (¬ 𝐴 = 𝐵 ↔ 𝜑))
4	1, 3	syl5bb 191	1 ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 819   = wceq 1331   ≠ wne 2308

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-dc 820  df-ne 2309

This theorem is referenced by:  necon2abiidc  2372