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Theorem sbcreu 3783
 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 3707 . 2 ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑𝐴 ∈ V)
2 reurex 3342 . . 3 (∃!𝑦𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)
3 sbcex 3707 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
43rexlimivw 3207 . . 3 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 (∃!𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V)
6 dfsbcq2 3700 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑[𝐴 / 𝑥]∃!𝑦𝐵 𝜑))
7 dfsbcq2 3700 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
87reubidv 3308 . . 3 (𝑧 = 𝐴 → (∃!𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
9 nfcv 2920 . . . . 5 𝑥𝐵
10 nfs1v 2158 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
119, 10nfreuw 3293 . . . 4 𝑥∃!𝑦𝐵 [𝑧 / 𝑥]𝜑
12 sbequ12 2251 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
1312reubidv 3308 . . . 4 (𝑥 = 𝑧 → (∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑))
1411, 13sbiev 2323 . . 3 ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑)
156, 8, 14vtoclbg 3488 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
161, 5, 15pm5.21nii 384 1 ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1539  [wsb 2070   ∈ wcel 2112  ∃wrex 3072  ∃!wreu 3073  Vcvv 3410  [wsbc 3697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-v 3412  df-sbc 3698 This theorem is referenced by: (None)
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