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Theorem cbvreuw 3389
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3394 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvreuw.1 𝑦𝜑
cbvreuw.2 𝑥𝜓
cbvreuw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreuw (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvreuw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . 4 𝑧(𝑥𝐴𝜑)
21sb8euv 2660 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑))
3 sban 2085 . . . 4 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
43eubii 2645 . . 3 (∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
5 clelsb3 2917 . . . . . 6 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
65anbi1i 626 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
76eubii 2645 . . . 4 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
8 nfv 1915 . . . . . 6 𝑦 𝑧𝐴
9 cbvreuw.1 . . . . . . 7 𝑦𝜑
109nfsbv 2338 . . . . . 6 𝑦[𝑧 / 𝑥]𝜑
118, 10nfan 1900 . . . . 5 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
12 nfv 1915 . . . . 5 𝑧(𝑦𝐴𝜓)
13 eleq1w 2872 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
14 sbequ 2088 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 cbvreuw.2 . . . . . . . 8 𝑥𝜓
16 cbvreuw.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16sbiev 2322 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜓)
1814, 17syl6bb 290 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
1913, 18anbi12d 633 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2011, 12, 19cbveuw 2666 . . . 4 (∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
217, 20bitri 278 . . 3 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
222, 4, 213bitri 300 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
23 df-reu 3113 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
24 df-reu 3113 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2522, 23, 243bitr4i 306 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wnf 1785  [wsb 2069  wcel 2111  ∃!weu 2628  ∃!wreu 3108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-10 2142  ax-11 2158  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clel 2870  df-reu 3113
This theorem is referenced by:  cbvrmowOLD  3391  cbvreuvw  3398  reu8nf  3806  poimirlem25  35082
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