Theorem wl-nfs1t 35623

Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2492. (Contributed by Wolf Lammen, 27-Jul-2019.)

Assertion

Ref	Expression
wl-nfs1t	⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem wl-nfs1t

Step	Hyp	Ref	Expression
1		sbequ12r 2248	. . . . . 6 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
2	1	equcoms 2024	. . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	2	sps 2180	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
4	3	drnf1 2443	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥[𝑦 / 𝑥]𝜑 ↔ Ⅎ𝑦𝜑))
5	4	biimprd 247	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑))
6		nfsb2 2487	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
7	6	a1d 25	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑))
8	5, 7	pm2.61i 182	1 ⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀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:  wl-sb8t  35634