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Theorem hbsbw 2344
 Description: Version of hbsb 2561 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 10-Jan-2024.)
Hypothesis
Ref Expression
hbsbw.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
hbsbw ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbsbw
StepHypRef Expression
1 hbsbw.1 . . . 4 (𝜑 → ∀𝑧𝜑)
21nf5i 2143 . . 3 𝑧𝜑
32nfsbv 2342 . 2 𝑧[𝑦 / 𝑥]𝜑
43nf5ri 2187 1 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1528  [wsb 2062 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 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-nf 1778  df-sb 2063 This theorem is referenced by:  hbab  2808  hblem  2941
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