Theorem ifpiddc 999

Description: Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifiddc 3641. (Contributed by BJ, 20-Sep-2019.)

Assertion

Ref	Expression
ifpiddc	⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))

Proof of Theorem ifpiddc

Step	Hyp	Ref	Expression
1		exmiddc 843	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		ifptru 997	. . 3 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
3		ifpfal 998	. . 3 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
4	2, 3	jaoi 723	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
5	1, 4	syl 14	1 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 715  DECID wdc 841  if-wif 985

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-in2 620  ax-io 716

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986

This theorem is referenced by: (None)