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Theorem hbsb3 2506
 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 2380. Check out bj-hbsb3v 34534 for a weaker version requiring less axioms. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)
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 2080 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3 hbsb2a 2503 . 2 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
42, 3syl 17 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1537  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by:  nfs1  2507  axc16ALT  2508
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