Theorem simplimdc 757

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		pm2.21 547	. 2 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
2		con1dc 753	. 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 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm2.5dc  763  dfandc  778  pm4.79dc  809