Theorem casesifp 1077

Description: Version of cases 1042 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 1074 and ifpfal 1075. (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 1042	. 2 ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))
4		df-ifp 1063	. 2 ⊢ (if-(𝜑, 𝜒, 𝜃) ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))
5	3, 4	bitr4i 278	1 ⊢ (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  if-wif 1062

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 207  df-an 396  df-or 848  df-ifp 1063

This theorem is referenced by:  hadifp  1605  cadifp  1619  brif1  7509  wl-1xor  37505  wl-1mintru1  37511  brif2  42242  brif12  42243