Theorem anifpdc 992

Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 993. (Contributed by BJ, 30-Sep-2019.)

Assertion

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

Proof of Theorem anifpdc

Step	Hyp	Ref	Expression
1		olc 716	. . 3 ⊢ (𝜓 → (¬ 𝜑 ∨ 𝜓))
2		olc 716	. . 3 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
3	1, 2	anim12i 338	. 2 ⊢ ((𝜓 ∧ 𝜒) → ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
4		dfifp4dc 989	. 2 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))))
5	3, 4	imbitrrid 156	1 ⊢ (DECID 𝜑 → ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)))

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-in1 617  ax-in2 618  ax-io 714

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

This theorem is referenced by: (None)