Theorem casesifp 1026

Description: Version of cases 992 expressed using if-. Case disjunction according to the value of 𝜑. One can see this as a proof that the two hypotheses characterize the conditional operator for propositions. For the converses, see ifptru 1023 and ifpfal 1024. (Contributed by BJ, 20-Sep-2019.)

Hypotheses

Ref	Expression
casesifp.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
casesifp.2	⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
casesifp	⊢ (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))

Proof of Theorem casesifp

Step	Hyp	Ref	Expression
1		casesifp.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		casesifp.2	. . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃))
3	1, 2	cases 992	. 2 ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))
4		df-ifp 1013	. 2 ⊢ (if-(𝜑, 𝜒, 𝜃) ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))
5	3, 4	bitr4i 267	1 ⊢ (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  if-wif 1012

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 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  hadifp  1544  cadifp  1557