Theorem ifpdfbidc 991

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
ifpdfbidc	⊢ (DECID 𝜑 → ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓)))

Proof of Theorem ifpdfbidc

Step	Hyp	Ref	Expression
1		con34bdc 876	. . 3 ⊢ (DECID 𝜑 → ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓)))
2	1	anbi2d 464	. 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓))))
3		dfbi2 388	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
4	3	a1i 9	. 2 ⊢ (DECID 𝜑 → ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
5		dfifp2dc 987	. 2 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, ¬ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓))))
6	2, 4, 5	3bitr4d 220	1 ⊢ (DECID 𝜑 → ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 839  if-wif 983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-ifp 984

This theorem is referenced by: (None)