Theorem simplimdc 855

Description: Simplification for a decidable proposition. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 29-Mar-2018.)

Assertion

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

Proof of Theorem simplimdc

Step	Hyp	Ref	Expression
1		pm2.21 612	. 2 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
2		con1dc 851	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → (𝜑 → 𝜓)) → (¬ (𝜑 → 𝜓) → 𝜑)))
3	1, 2	mpi 15	1 ⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 829

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-stab 826  df-dc 830

This theorem is referenced by:  pm2.5gdc  861  dfandc  879  pm4.79dc  898