Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcralg Structured version   Visualization version   GIF version

Theorem sbcralg 3861
 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 2982 . 2 𝑦𝐴
2 sbcralt 3859 . 2 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
31, 2mpan2 687 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2107  Ⅎwnfc 2966  ∀wral 3143  [wsbc 3776 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-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-v 3502  df-sbc 3777 This theorem is referenced by:  r19.12sn  4655  bnj538  31897  csbwrecsg  34477  cdlemkid3N  37936  cdlemkid4  37937  rspsbc2  40733  rspsbc2VD  41054
 Copyright terms: Public domain W3C validator