Theorem ifpdfbiOLD 1087

Description: Obsolete version of ifpdfbi 1086 as of 25-Jun-2026. Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ifpdfbiOLD	⊢ ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

Proof of Theorem ifpdfbiOLD

Step	Hyp	Ref	Expression
1		con34b 319	. . 3 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
2	1	anbi2i 634	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
3		dfbi2 479	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
4		dfifp2 1080	. 2 ⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
5	2, 3, 4	3bitr4i 306	1 ⊢ ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  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 401  df-or 861  df-ifp 1079

This theorem is used by: (None)