Theorem necon1bbiidc 2401

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

Hypothesis

Ref	Expression
necon1bbiidc.1	⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜑))

Assertion

Ref	Expression
necon1bbiidc	⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 ↔ 𝐴 = 𝐵))

Proof of Theorem necon1bbiidc

Step	Hyp	Ref	Expression
1		df-ne 2341	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1bbiidc.1	. . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜑))
3	1, 2	bitr3id 193	. 2 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 ↔ 𝜑))
4	3	con1biidc 872	1 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 829   = wceq 1348   ≠ wne 2340

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830  df-ne 2341

This theorem is referenced by:  necon2bbiidc  2405