Theorem ifpdfbi 1081

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

Assertion

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

Proof of Theorem ifpdfbi

Step	Hyp	Ref	Expression
1		con34b 318	. . 3 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
2	1	anbi2i 632	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
3		dfbi2 478	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
4		dfifp2 1075	. 2 ⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
5	2, 3, 4	3bitr4i 305	1 ⊢ ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  if-wif 1073

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 209  df-an 400  df-or 859  df-ifp 1074

This theorem is referenced by:  wl-df3xor2  37923  wl-2xor  37937  ifpbiidcor  44010  ifpbicor  44011