Theorem ifpimpda 1022

Description: Separation of the values of the conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 27-Feb-2021.)

Hypotheses

Ref	Expression
ifpimpda.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
ifpimpda.2	⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃)

Assertion

Ref	Expression
ifpimpda	⊢ (𝜑 → if-(𝜓, 𝜒, 𝜃))

Proof of Theorem ifpimpda

Step	Hyp	Ref	Expression
1		ifpimpda.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 449	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		ifpimpda.2	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃)
4	3	ex 449	. 2 ⊢ (𝜑 → (¬ 𝜓 → 𝜃))
5		dfifp2 1008	. 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 → 𝜒) ∧ (¬ 𝜓 → 𝜃)))
6	2, 4, 5	sylanbrc 695	1 ⊢ (𝜑 → if-(𝜓, 𝜒, 𝜃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ 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:  ifpprsnss  40845  1wlkp1lem8  40889  11wlkdlem4  41307