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Theorem sbcex2 3095
 Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
Assertion
Ref Expression
sbcex2 ([̣A / yxφxA / yφ)
Distinct variable groups:   x,A   x,y
Allowed substitution hints:   φ(x,y)   A(y)

Proof of Theorem sbcex2
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbcex 3055 . 2 ([̣A / yxφA V)
2 sbcex 3055 . . 3 ([̣A / yφA V)
32exlimiv 1634 . 2 (xA / yφA V)
4 dfsbcq2 3049 . . 3 (z = A → ([z / y]xφ ↔ [̣A / yxφ))
5 dfsbcq2 3049 . . . 4 (z = A → ([z / y]φ ↔ [̣A / yφ))
65exbidv 1626 . . 3 (z = A → (x[z / y]φxA / yφ))
7 sbex 2128 . . 3 ([z / y]xφx[z / y]φ)
84, 6, 7vtoclbg 2915 . 2 (A V → ([̣A / yxφxA / yφ))
91, 3, 8pm5.21nii 342 1 ([̣A / yxφxA / yφ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  Vcvv 2859  [̣wsbc 3046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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: (None)
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