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Theorem sbcreu 3884
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 3800 . 2 ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑𝐴 ∈ V)
2 reurex 3381 . . 3 (∃!𝑦𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)
3 sbcex 3800 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
43rexlimivw 3148 . . 3 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 (∃!𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V)
6 dfsbcq2 3793 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑[𝐴 / 𝑥]∃!𝑦𝐵 𝜑))
7 dfsbcq2 3793 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
87reubidv 3395 . . 3 (𝑧 = 𝐴 → (∃!𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
9 nfcv 2902 . . . . 5 𝑥𝐵
10 nfs1v 2153 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
119, 10nfreuw 3411 . . . 4 𝑥∃!𝑦𝐵 [𝑧 / 𝑥]𝜑
12 sbequ12 2248 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
1312reubidv 3395 . . . 4 (𝑥 = 𝑧 → (∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑))
1411, 13sbiev 2312 . . 3 ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑)
156, 8, 14vtoclbg 3556 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
161, 5, 15pm5.21nii 378 1 ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  [wsb 2061  wcel 2105  wrex 3067  ∃!wreu 3375  Vcvv 3477  [wsbc 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-rex 3068  df-rmo 3377  df-reu 3378  df-v 3479  df-sbc 3791
This theorem is referenced by: (None)
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