Theorem ifpid2 40976

Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpid2	⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥))

Proof of Theorem ifpid2

Step	Hyp	Ref	Expression
1		tru 1543	. . . 4 ⊢ ⊤
2	1	olci 862	. . 3 ⊢ (¬ 𝜑 ∨ ⊤)
3	2	biantrur 530	. 2 ⊢ ((𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ ⊥)))
4		fal 1553	. . 3 ⊢ ¬ ⊥
5	4	biorfi 935	. 2 ⊢ (𝜑 ↔ (𝜑 ∨ ⊥))
6		dfifp4 1063	. 2 ⊢ (if-(𝜑, ⊤, ⊥) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ ⊥)))
7	3, 5, 6	3bitr4i 302	1 ⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 843  if-wif 1059  ⊤wtru 1540  ⊥wfal 1551

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 206  df-an 396  df-or 844  df-ifp 1060  df-tru 1542  df-fal 1552

This theorem is referenced by:  frege52aid  41355