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Theorem sbcexg 3096
 Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
Assertion
Ref Expression
sbcexg (A V → ([̣A / yxφxA / yφ))
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 3049 . 2 (z = A → ([z / y]xφ ↔ [̣A / yxφ))
2 dfsbcq2 3049 . . 3 (z = A → ([z / y]φ ↔ [̣A / yφ))
32exbidv 1626 . 2 (z = A → (x[z / y]φxA / yφ))
4 sbex 2128 . 2 ([z / y]xφx[z / y]φ)
51, 3, 4vtoclbg 2915 1 (A V → ([̣A / yxφxA / yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣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-v 2861  df-sbc 3047 This theorem is referenced by:  sbcabel  3123  csbunig  3899  csbxpg  4813  csbrng  4966
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