Theorem anifp 1014

Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1015. (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 398	. . 3 ⊢ (𝜓 → (¬ 𝜑 ∨ 𝜓))
2		olc 398	. . 3 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
3	1, 2	anim12i 588	. 2 ⊢ ((𝜓 ∧ 𝜒) → ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
4		dfifp4 1010	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	3, 4	sylibr 223	1 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383  if-wif 1006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by:  bj-consensus  31732  bj-consensusALT  31733  axfrege58a  37188