Theorem ifpdfan2 37807

Description: Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpdfan2	⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑))

Proof of Theorem ifpdfan2

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
2	1	notnoti 137	. . 3 ⊢ ¬ ¬ (𝜑 → 𝜑)
3	2	biorfi 422	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 → 𝜑)))
4		dfifp6 1018	. 2 ⊢ (if-(𝜑, 𝜓, 𝜑) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 → 𝜑)))
5	3, 4	bitr4i 267	1 ⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  if-wif 1012

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 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  ifpancor  37808