Theorem ifpdc 985

Description: The conditional operator for propositions implies decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)

Assertion

Ref	Expression
ifpdc	⊢ (if-(𝜑, 𝜓, 𝜒) → DECID 𝜑)

Proof of Theorem ifpdc

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2		simpl 109	. . 3 ⊢ ((¬ 𝜑 ∧ 𝜒) → ¬ 𝜑)
3	1, 2	orim12i 764	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → (𝜑 ∨ ¬ 𝜑))
4		df-ifp 984	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
5		df-dc 840	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
6	3, 4, 5	3imtr4i 201	1 ⊢ (if-(𝜑, 𝜓, 𝜒) → DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713  DECID wdc 839  if-wif 983

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-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984

This theorem is referenced by:  ifpnst  994