Theorem ifpancor 44231

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 466	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		ifpdfan2 44230	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑))
3		ifpdfan2 44230	. 2 ⊢ ((𝜓 ∧ 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))
4	1, 2, 3	3bitr3i 304	1 ⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  ifpnancor  44248