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Theorem sbcreu 3721
Description: Interchange class substitution and restricted unique existential 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 3654 . 2 ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑𝐴 ∈ V)
2 reurex 3360 . . 3 (∃!𝑦𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)
3 sbcex 3654 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
43rexlimivw 3228 . . 3 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 (∃!𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V)
6 dfsbcq2 3647 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑[𝐴 / 𝑥]∃!𝑦𝐵 𝜑))
7 dfsbcq2 3647 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
87reubidv 3326 . . 3 (𝑧 = 𝐴 → (∃!𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
9 nfcv 2959 . . . . 5 𝑥𝐵
10 nfs1v 2288 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
119, 10nfreu 3313 . . . 4 𝑥∃!𝑦𝐵 [𝑧 / 𝑥]𝜑
12 sbequ12 2280 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
1312reubidv 3326 . . . 4 (𝑥 = 𝑧 → (∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑))
1411, 13sbie 2569 . . 3 ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑)
156, 8, 14vtoclbg 3471 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
161, 5, 15pm5.21nii 369 1 ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  [wsb 2061  wcel 2157  wrex 3108  ∃!wreu 3109  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-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-v 3404  df-sbc 3645
This theorem is referenced by: (None)
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