Theorem ifporcor 40170

Description: Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)

Assertion

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

Proof of Theorem ifporcor

Step	Hyp	Ref	Expression
1		orcom 867	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
2		ifpdfor2 40169	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓))
3		ifpdfor2 40169	. 2 ⊢ ((𝜓 ∨ 𝜑) ↔ if-(𝜓, 𝜓, 𝜑))
4	1, 2, 3	3bitr3i 304	1 ⊢ (if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑))

Syntax hints:   ↔ wb 209   ∨ wo 844  if-wif 1058

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 210  df-an 400  df-or 845  df-ifp 1059

This theorem is referenced by:  ifpnorcor  40188