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Theorem sbcralg 3120
 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 (A V → ([̣A / xy B φy BA / xφ))
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
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . 2 (z = A → ([z / x]y B φ ↔ [̣A / xy B φ))
2 dfsbcq2 3049 . . 3 (z = A → ([z / x]φ ↔ [̣A / xφ))
32ralbidv 2634 . 2 (z = A → (y B [z / x]φy BA / xφ))
4 nfcv 2489 . . . 4 xB
5 nfs1v 2106 . . . 4 x[z / x]φ
64, 5nfral 2667 . . 3 xy B [z / x]φ
7 sbequ12 1919 . . . 4 (x = z → (φ ↔ [z / x]φ))
87ralbidv 2634 . . 3 (x = z → (y B φy B [z / x]φ))
96, 8sbie 2038 . 2 ([z / x]y B φy B [z / x]φ)
101, 3, 9vtoclbg 2915 1 (A V → ([̣A / xy B φy BA / xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∀wral 2614  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047 This theorem is referenced by:  r19.12sn  3789
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