Theorem ifpdfbi 1084

Description: Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.) (Proof shortened by Garrett Katz, 25-Jun-2026.)

Assertion

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

Proof of Theorem ifpdfbi

Step	Hyp	Ref	Expression
1		dfbi3 1063	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2		df-ifp 1077	. 2 ⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
3	1, 2	bitr4i 281	1 ⊢ ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860  if-wif 1076

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 210  df-an 401  df-or 861  df-ifp 1077

This theorem is referenced by:  wl-df3xor2  37970  wl-2xor  37984  ifpbiidcor  44057  ifpbicor  44058