Theorem necon1aidc 2398

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon1aidc.1	⊢ (DECID 𝜑 → (¬ 𝜑 → 𝐴 = 𝐵))

Assertion

Ref	Expression
necon1aidc	⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 → 𝜑))

Proof of Theorem necon1aidc

Step	Hyp	Ref	Expression
1		df-ne 2348	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1aidc.1	. . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 → 𝐴 = 𝐵))
3		con1dc 856	. . 3 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝐴 = 𝐵) → (¬ 𝐴 = 𝐵 → 𝜑)))
4	2, 3	mpd 13	. 2 ⊢ (DECID 𝜑 → (¬ 𝐴 = 𝐵 → 𝜑))
5	1, 4	biimtrid 152	1 ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 834   = wceq 1353   ≠ wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-ne 2348

This theorem is referenced by:  necon1idc  2400  lgsne0  14600