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Theorem sbcralg 2809
 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 / x]y B φy B [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
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2740 . 2 (z = A → ([z / x]y B φ[A / x]y B φ))
2 dfsbcq2 2740 . . 3 (z = A → ([z / x]φ[A / x]φ))
32ralbidv 2300 . 2 (z = A → (y B [z / x]φy B [A / x]φ))
4 nfcv 2156 . . . 4 xB
5 nfs1v 1793 . . . 4 x[z / x]φ
64, 5nfralxy 2334 . . 3 xy B [z / x]φ
7 sbequ12 1632 . . . 4 (x = z → (φ ↔ [z / x]φ))
87ralbidv 2300 . . 3 (x = z → (y B φy B [z / x]φ))
96, 8sbie 1652 . 2 ([z / x]y B φy B [z / x]φ)
101, 3, 9vtoclbg 2587 1 (A 𝑉 → ([A / x]y B φy B [A / x]φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1226   ∈ wcel 1370  [wsb 1623  ∀wral 2280  [wsbc 2737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-sbc 2738 This theorem is referenced by:  r19.12sn  3406
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