Theorem anifp 1088

Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1089. (Contributed by BJ, 30-Sep-2019.)

Assertion

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

Proof of Theorem anifp

Step	Hyp	Ref	Expression
1		olc 882	. . 3 ⊢ (𝜓 → (¬ 𝜑 ∨ 𝜓))
2		olc 882	. . 3 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
3	1, 2	anim12i 625	. 2 ⊢ ((𝜓 ∧ 𝜒) → ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
4		dfifp4 1082	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	3, 4	sylibr 237	1 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  psdmvr  22490  bj-consensus  37448  bj-consensusALT  37449  axfrege58a  44874