Theorem simprimdc 849

Description: Simplification given a decidable proposition. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 30-Apr-2018.)

Assertion

Ref	Expression
simprimdc	⊢ (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) → 𝜓))

Proof of Theorem simprimdc

Step	Hyp	Ref	Expression
1		idd 21	. . 3 ⊢ (𝜑 → (𝜓 → 𝜓))
2	1	a1i 9	. 2 ⊢ (DECID 𝜓 → (𝜑 → (𝜓 → 𝜓)))
3	2	impidc 848	1 ⊢ (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 824

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  dfandc  874