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Theorem bj-sb3v 32731
Description: Version of sb3 2353 with a dv condition, which does not require ax-13 2244. This allows to remove ax-13 2244 from sb5 2428 (see bj-sb5 32743). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sb3v (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-sb3v
StepHypRef Expression
1 sb56 2148 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2 bj-sb2v 32728 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
31, 2sylbi 207 1 (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1479  wex 1702  [wsb 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708  df-sb 1879
This theorem is referenced by: (None)
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