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Theorem sb2v 2302
 Description: Version of sb2 2481 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by BJ, 31-May-2019.)
Assertion
Ref Expression
sb2v (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb2v
StepHypRef Expression
1 sp 2226 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 equs4v 2109 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 2070 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 580 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1656  ∃wex 1880  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-12 2222 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-sb 2070 This theorem is referenced by:  stdpc4v  2303  equsb1v  2306  sb6  2309  sbi1v  2337  iota4  6103  bj-sb3v  33286  bj-hbs1  33287  bj-hbsb2av  33289  wl-lem-moexsb  33893  absnsb  41962
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