Theorem casesifp 1094

Description: Version of cases 1058 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 1091 and ifpfal 1092. (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 1058	. 2 ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))
4		df-ifp 1079	. 2 ⊢ (if-(𝜑, 𝜒, 𝜃) ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))
5	3, 4	bitr4i 281	1 ⊢ (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ 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:  hadifp  1637  hadifpOLD  1638  cadifp  1652  brif1  7513  wl-1xor  38238  wl-1mintru1  38244  brif2  43096  brif12  43097