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Theorem sbcex2 3856
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 3801 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 3801 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32exlimiv 1928 . 2 (∃𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 3794 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 3794 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65exbidv 1919 . . 3 (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
7 sbex 2280 . . 3 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 3557 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 378 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wex 1776  [wsb 2062  wcel 2106  Vcvv 3478  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sbc 3792
This theorem is referenced by:  sbcabel  3887  csbuni  4941  csbxp  5788  csbdm  5911  sbcfung  6592  csbfrecsg  8308  bnj89  34714  bnj985v  34946  bnj985  34947  csboprabg  37313  sbcexf  38102  onfrALTlem5  44540  onfrALTlem5VD  44883  csbxpgVD  44892  csbrngVD  44894  csbunigVD  44896
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