Theorem bj-nfs1t 33316

Description: A theorem close to a closed form of nfs1 2440. (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 33314	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
2	1	axc4i 2296	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
3		nf5 2255	. 2 ⊢ (Ⅎ𝑥[𝑦 / 𝑥]𝜑 ↔ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
4	2, 3	sylibr 226	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1599  Ⅎwnf 1827  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162  ax-13 2333

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012

This theorem is referenced by:  bj-nfs1t2  33317