Theorem bj-nfs1t 37453

Description: A theorem close to a closed form of nfs1 2519. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-nfs1t	⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem bj-nfs1t

Step	Hyp	Ref	Expression
1		bj-hbsb3t 37451	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
2	1	axc4i 2354	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
3		nf5 2316	. 2 ⊢ (Ⅎ𝑥[𝑦 / 𝑥]𝜑 ↔ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
4	2, 3	sylibr 237	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  Ⅎwnf 1812  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-nf 1813  df-sb 2096

This theorem is used by:  bj-nfs1t2  37454