Theorem ifpdfan2 39821

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 145	. . 3 ⊢ ¬ ¬ (𝜑 → 𝜑)
3	2	biorfi 935	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 → 𝜑)))
4		dfifp6 1063	. 2 ⊢ (if-(𝜑, 𝜓, 𝜑) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 → 𝜑)))
5	3, 4	bitr4i 280	1 ⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843  if-wif 1057

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 399  df-or 844  df-ifp 1058

This theorem is referenced by:  ifpancor  39822