Theorem bj-nfs1t2 36177

Description: A theorem close to a closed form of nfs1 2481. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-nfs1t2	⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem bj-nfs1t2

Step	Hyp	Ref	Expression
1		nf5r 2179	. . 3 ⊢ (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
2	1	alimi 1805	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
3		bj-nfs1t 36176	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
4	2, 3	syl 17	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1531  Ⅎwnf 1777  [wsb 2059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2365

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060

This theorem is referenced by:  bj-nfs1  36178