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Theorem sbcexg 2807
 Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
Assertion
Ref Expression
sbcexg (A 𝑉 → ([A / y]xφx[A / y]φ))
Distinct variable groups:   x,A   x,y
Allowed substitution hints:   φ(x,y)   A(y)   𝑉(x,y)

Proof of Theorem sbcexg
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . 2 (z = A → ([z / y]xφ[A / y]xφ))
2 dfsbcq2 2761 . . 3 (z = A → ([z / y]φ[A / y]φ))
32exbidv 1703 . 2 (z = A → (x[z / y]φx[A / y]φ))
4 sbex 1877 . 2 ([z / y]xφx[z / y]φ)
51, 3, 4vtoclbg 2608 1 (A 𝑉 → ([A / y]xφx[A / y]φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  [wsb 1642  [wsbc 2758 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759 This theorem is referenced by:  sbcabel  2833  csbunig  3579  csbxpg  4364  csbrng  4725
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