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Theorem wl-nfs1t 33656
Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2503. (Contributed by Wolf Lammen, 27-Jul-2019.)
Assertion
Ref Expression
wl-nfs1t (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem wl-nfs1t
StepHypRef Expression
1 sbequ12r 2260 . . . . . 6 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
21equcoms 2103 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
32sps 2203 . . . 4 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
43drnf1 2470 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥[𝑦 / 𝑥]𝜑 ↔ Ⅎ𝑦𝜑))
54biimprd 238 . 2 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑))
6 nfsb2 2498 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
76a1d 25 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑))
85, 7pm2.61i 176 1 (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1630  wnf 1857  [wsb 2047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-10 2169  ax-12 2197  ax-13 2392
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-sb 2048
This theorem is referenced by:  wl-sb8t  33665
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