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Theorem nfs1 2353
Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfs1.1 𝑦𝜑
Assertion
Ref Expression
nfs1 𝑥[𝑦 / 𝑥]𝜑

Proof of Theorem nfs1
StepHypRef Expression
1 nfs1.1 . . . 4 𝑦𝜑
21nf5ri 2053 . . 3 (𝜑 → ∀𝑦𝜑)
32hbsb3 2352 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
43nf5i 2011 1 𝑥[𝑦 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wnf 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:  sb8  2412  sb8e  2413
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