Theorem wl-sbnf1

Description: Two ways expressing that x is effectively not free in ph . Simplified version of sbnf2 . Note: This theorem shows that sbnf2 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019)

	Ref	Expression
Assertion	wl-sbnf1	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nf5	⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) )
2		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 Ⅎ 𝑦 𝜑
3		wl-sbhbt	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑦 ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )
4	2 3	albid	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )
5	1 4	bitrid	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )