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Theorem nfs1 2479
Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Usage of this theorem is discouraged because it depends on ax-13 2363. Check out nfs1v 2145 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
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 2180 . . 3 (𝜑 → ∀𝑦𝜑)
32hbsb3 2478 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
43nf5i 2134 1 𝑥[𝑦 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wnf 1777  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2363
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060
This theorem is referenced by:  sb8  2508  sb8e  2509
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