Theorem bj-nfs1t2 34900

Description: A theorem close to a closed form of nfs1 2492. (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 2189	. . 3 ⊢ (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
2	1	alimi 1815	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
3		bj-nfs1t 34899	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
4	2, 3	syl 17	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1537  Ⅎwnf 1787  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-sb 2069

This theorem is referenced by:  bj-nfs1  34901