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Theorem sbcex2 3695
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 3654 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 3654 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32exlimiv 2021 . 2 (∃𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 3647 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 3647 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65exbidv 2012 . . 3 (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
7 sbex 2626 . . 3 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 3471 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 369 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wex 1859  [wsb 2061  wcel 2157  Vcvv 3402  [wsbc 3644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-v 3404  df-sbc 3645
This theorem is referenced by:  sbcabel  3723  csbuni  4671  csbxp  5413  csbdm  5530  sbcfung  6132  bnj89  31122  bnj985  31355  csbwrecsg  33496  csboprabg  33499  sbcexf  34235  onfrALTlem5  39260  csbxpgOLD  39553  csbrngOLD  39556  onfrALTlem5VD  39620
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