Theorem necon2abiidc 2370

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

Hypothesis

Ref	Expression
necon2abiidc.1	⊢ (DECID 𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜑))

Assertion

Ref	Expression
necon2abiidc	⊢ (DECID 𝜑 → (𝜑 ↔ 𝐴 ≠ 𝐵))

Proof of Theorem necon2abiidc

Step	Hyp	Ref	Expression
1		necon2abiidc.1	. . . 4 ⊢ (DECID 𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜑))
2	1	bicomd 140	. . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵))
3	2	necon1abiidc 2366	. 2 ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑))
4	3	bicomd 140	1 ⊢ (DECID 𝜑 → (𝜑 ↔ 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-dc 820  df-ne 2307

This theorem is referenced by: (None)