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Theorem hbsb3 1819
Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
hbsb3.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
hbsb3 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem hbsb3
StepHypRef Expression
1 hbsb3.1 . . 3 (𝜑 → ∀𝑦𝜑)
21sbimi 1775 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3 hbsb2a 1817 . 2 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
42, 3syl 14 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-11 1517  ax-4 1521  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-sb 1774
This theorem is referenced by:  nfs1  1820  sbcof2  1821  ax16  1824  sb8h  1865  sb8eh  1866  ax16ALT  1870
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