Theorem dfifp3dc 988

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
dfifp3dc	⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))))

Proof of Theorem dfifp3dc

Step	Hyp	Ref	Expression
1		dfifp2dc 987	. 2 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))))
2		pm4.64dc 905	. . 3 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜒) ↔ (𝜑 ∨ 𝜒)))
3	2	anbi2d 464	. 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))))
4	1, 3	bitrd 188	1 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ 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:  dfifp4dc  989  ifpnst  994