Theorem nanbi1OLD 1623

Description: Obsolete proof of nanbi1 1622 as of 19-Oct-2022. (Contributed by Anthony Hart, 1-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nanbi1OLD	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)))

Proof of Theorem nanbi1OLD

Step	Hyp	Ref	Expression
1		anbi1 626	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)))
2	1	notbid 310	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ (𝜑 ∧ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒)))
3		df-nan 1610	. 2 ⊢ ((𝜑 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜒))
4		df-nan 1610	. 2 ⊢ ((𝜓 ⊼ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒))
5	2, 3, 4	3bitr4g 306	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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)