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Theorem nfs1 2364
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 2064 . . 3 (𝜑 → ∀𝑦𝜑)
32hbsb3 2363 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
43nf5i 2023 1 𝑥[𝑦 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wnf 1707  [wsb 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-12 2046  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1704  df-nf 1709  df-sb 1880
This theorem is referenced by:  sb8  2423  sb8e  2424
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