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Theorem sbcex2 3468
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 3427 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 3427 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32exlimiv 1855 . 2 (∃𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 3420 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 3420 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65exbidv 1847 . . 3 (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
7 sbex 2462 . . 3 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 3253 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 368 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wex 1701  [wsb 1877  wcel 1987  Vcvv 3186  [wsbc 3417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188  df-sbc 3418
This theorem is referenced by:  sbcabel  3498  csbuni  4432  csbxp  5161  csbdm  5278  sbcfung  5871  bnj89  30492  bnj985  30728  csbwrecsg  32802  csboprabg  32805  sbcexf  33547  onfrALTlem5  38236
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