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Theorem sbcreug 2895
Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
sbcreug (𝐴𝑉 → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcreug
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2819 . 2 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑[𝐴 / 𝑥]∃!𝑦𝐵 𝜑))
2 dfsbcq2 2819 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32reubidv 2538 . 2 (𝑧 = 𝐴 → (∃!𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
4 nfcv 2220 . . . 4 𝑥𝐵
5 nfs1v 1857 . . . 4 𝑥[𝑧 / 𝑥]𝜑
64, 5nfreuxy 2529 . . 3 𝑥∃!𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 1695 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87reubidv 2538 . . 3 (𝑥 = 𝑧 → (∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 1715 . 2 ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑)
101, 3, 9vtoclbg 2660 1 (𝐴𝑉 → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  [wsb 1686  ∃!wreu 2351  [wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-reu 2356  df-v 2604  df-sbc 2817
This theorem is referenced by: (None)
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