Theorem bj-nfs1t 37280

Description: A theorem close to a closed form of nfs1 2520. (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 37278	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
2	1	axc4i 2355	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
3		nf5 2317	. 2 ⊢ (Ⅎ𝑥[𝑦 / 𝑥]𝜑 ↔ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
4	2, 3	sylibr 236	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1559  Ⅎwnf 1804  [wsb 2091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-10 2176  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1801  df-nf 1805  df-sb 2092

This theorem is referenced by:  bj-nfs1t2  37281