Theorem nancomOLD 1615

Description: Obsolete proof of nancom 1614 as of 19-Oct-2022. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nancomOLD	⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑))

Proof of Theorem nancomOLD

Step	Hyp	Ref	Expression
1		df-nan 1610	. . 3 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2		ancom 453	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
3	1, 2	xchbinx 326	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜓 ∧ 𝜑))
4		df-nan 1610	. 2 ⊢ ((𝜓 ⊼ 𝜑) ↔ ¬ (𝜓 ∧ 𝜑))
5	3, 4	bitr4i 270	1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 385   ⊼ wnan 1609

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 199  df-an 386  df-nan 1610

This theorem is referenced by: (None)