Theorem ifpancor 44004

Description: Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpancor	⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))

Proof of Theorem ifpancor

Step	Hyp	Ref	Expression
1		ancom 464	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		ifpdfan2 44003	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑))
3		ifpdfan2 44003	. 2 ⊢ ((𝜓 ∧ 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))
4	1, 2, 3	3bitr3i 303	1 ⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 399  if-wif 1073

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 400  df-or 859  df-ifp 1074

This theorem is referenced by:  ifpnancor  44021