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Theorem sbcex2 3638
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcex2 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
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 3597 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 3597 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32exlimiv 2010 . 2 (∃𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 3590 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 3590 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65exbidv 2002 . . 3 (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
7 sbex 2611 . . 3 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 3418 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 367 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wex 1852  [wsb 2049  wcel 2145  Vcvv 3351  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by:  sbcabel  3666  csbuni  4603  csbxp  5339  csbdm  5455  sbcfung  6054  bnj89  31127  bnj985  31361  csbwrecsg  33510  csboprabg  33513  sbcexf  34250  onfrALTlem5  39282  csbxpgOLD  39576  csbrngOLD  39579  onfrALTlem5VD  39643
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