Theorem ifpdfan 41286

Description: Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpdfan	⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, ⊥))

Proof of Theorem ifpdfan

Step	Hyp	Ref	Expression
1		fal 1553	. . . 4 ⊢ ¬ ⊥
2	1	intnan 488	. . 3 ⊢ ¬ (¬ 𝜑 ∧ ⊥)
3	2	biorfi 937	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ⊥)))
4		df-ifp 1062	. 2 ⊢ (if-(𝜑, 𝜓, ⊥) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ⊥)))
5	3, 4	bitr4i 278	1 ⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, ⊥))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   ∨ wo 845  if-wif 1061  ⊥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 398  df-or 846  df-ifp 1062  df-tru 1542  df-fal 1552

This theorem is referenced by:  ifpdfnan  41306  ifpdfxor  41307