Theorem bj-nfs1t2 37795

Description: A theorem close to a closed form of nfs1 2517. (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 2230	. . 3 ⊢ (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
2	1	alimi 1843	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
3		bj-nfs1t 37794	. 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 1567  Ⅎwnf 1815  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099

This theorem is used by:  bj-nfs1  37796