Theorem bj-nfs1t2 37467

Description: A theorem close to a closed form of nfs1 2523. (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 2233	. . 3 ⊢ (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
2	1	alimi 1844	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
3		bj-nfs1t 37466	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
4	2, 3	syl 18	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  bj-nfs1  37468