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Theorem sbcralg 3671
Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcralg (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcralg
StepHypRef Expression
1 nfcv 2907 . 2 𝑦𝐴
2 sbcralt 3669 . 2 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
31, 2mpan2 682 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wcel 2155  wnfc 2894  wral 3055  [wsbc 3596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-sbc 3597
This theorem is referenced by:  r19.12sn  4410  bnj538  31258  csbwrecsg  33607  cdlemkid3N  36889  cdlemkid4  36890  rspsbc2  39412  rspsbc2VD  39743
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