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Theorem hbsb3 1689
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 1647 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3 hbsb2a 1687 . 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 1241  [wsb 1645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-11 1397  ax-4 1400  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-sb 1646
This theorem is referenced by:  nfs1  1690  sbcof2  1691  ax16  1694  sb8h  1734  sb8eh  1735  ax16ALT  1739
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