Theorem necon2abiidc 2309

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 139	. . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵))
3	2	necon1abiidc 2305	. 2 ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑))
4	3	bicomd 139	1 ⊢ (DECID 𝜑 → (𝜑 ↔ 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)