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Theorem sbcex2 2812
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
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 2772 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 2772 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32exlimiv 1489 . 2 (∃𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 2767 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 2767 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65exbidv 1706 . . 3 (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
7 sbex 1880 . . 3 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 2614 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 620 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1243  wex 1381  wcel 1393  [wsb 1645  Vcvv 2557  [wsbc 2764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765
This theorem is referenced by:  csbdmg  4529
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