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Theorem sbcreu 3513
Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcreu ([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)

Proof of Theorem sbcreu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3443 . 2 ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑𝐴 ∈ V)
2 reurex 3158 . . 3 (∃!𝑦𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)
3 sbcex 3443 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
43rexlimivw 3027 . . 3 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 (∃!𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V)
6 dfsbcq2 3436 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑[𝐴 / 𝑥]∃!𝑦𝐵 𝜑))
7 dfsbcq2 3436 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
87reubidv 3124 . . 3 (𝑧 = 𝐴 → (∃!𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
9 nfcv 2763 . . . . 5 𝑥𝐵
10 nfs1v 2436 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
119, 10nfreu 3112 . . . 4 𝑥∃!𝑦𝐵 [𝑧 / 𝑥]𝜑
12 sbequ12 2110 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
1312reubidv 3124 . . . 4 (𝑥 = 𝑧 → (∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑))
1411, 13sbie 2407 . . 3 ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑)
156, 8, 14vtoclbg 3265 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
161, 5, 15pm5.21nii 368 1 ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1482  [wsb 1879  wcel 1989  wrex 2912  ∃!wreu 2913  Vcvv 3198  [wsbc 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-v 3200  df-sbc 3434
This theorem is referenced by: (None)
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