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Theorem nfs1 2502
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 2212 . . 3 (𝜑 → ∀𝑦𝜑)
32hbsb3 2501 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
43nf5i 2173 1 𝑥[𝑦 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wnf 1857  [wsb 2046
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 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-sb 2047
This theorem is referenced by:  sb8  2561  sb8e  2562
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