Theorem ifpancor 36827

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 465	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		ifpdfan2 36826	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑))
3		ifpdfan2 36826	. 2 ⊢ ((𝜓 ∧ 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))
4	1, 2, 3	3bitr3i 289	1 ⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))

Syntax hints:   ↔ wb 195   ∧ wa 383  if-wif 1006

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 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by:  ifpnancor  36845