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Theorem stdpc4v 2244
 Description: If a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑦 (properly) substituted for 𝑥. Version of stdpc4 2428 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 31-May-2019.)
Assertion
Ref Expression
stdpc4v (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem stdpc4v
StepHypRef Expression
1 ala1 1857 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
2 sb2v 2243 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
31, 2syl 17 1 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1599  [wsb 2011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-sb 2012 This theorem is referenced by:  sbftv  2245  spsbimvOLD  2284  pm13.183  3549  bj-2stdpc4v  33346  bj-sbfvv  33352  bj-sbtv  33353  bj-vexwvt  33433  bj-ab0  33481
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