Theorem wl-nfs1t 32503

Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2353. (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 2098	. . . . . 6 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
2	1	equcoms 1934	. . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	2	sps 2043	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
4	3	drnf1 2317	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥[𝑦 / 𝑥]𝜑 ↔ Ⅎ𝑦𝜑))
5	4	biimprd 237	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑))
6		nfsb2 2348	. . 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 175	1 ⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by:  wl-sb8t  32512