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Theorem cbvreu 3161
Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
cbvral.1 𝑦𝜑
cbvral.2 𝑥𝜓
cbvral.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreu (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvreu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1840 . . . 4 𝑧(𝑥𝐴𝜑)
21sb8eu 2502 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑))
3 sban 2398 . . . 4 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
43eubii 2491 . . 3 (∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
5 clelsb3 2726 . . . . . 6 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
65anbi1i 730 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
76eubii 2491 . . . 4 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
8 nfv 1840 . . . . . 6 𝑦 𝑧𝐴
9 cbvral.1 . . . . . . 7 𝑦𝜑
109nfsb 2439 . . . . . 6 𝑦[𝑧 / 𝑥]𝜑
118, 10nfan 1825 . . . . 5 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
12 nfv 1840 . . . . 5 𝑧(𝑦𝐴𝜓)
13 eleq1 2686 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
14 sbequ 2375 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 cbvral.2 . . . . . . . 8 𝑥𝜓
16 cbvral.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16sbie 2407 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜓)
1814, 17syl6bb 276 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
1913, 18anbi12d 746 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2011, 12, 19cbveu 2504 . . . 4 (∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
217, 20bitri 264 . . 3 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
222, 4, 213bitri 286 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
23 df-reu 2915 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
24 df-reu 2915 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2522, 23, 243bitr4i 292 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wnf 1705  [wsb 1877  wcel 1987  ∃!weu 2469  ∃!wreu 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-cleq 2614  df-clel 2617  df-reu 2915
This theorem is referenced by:  cbvrmo  3162  cbvreuv  3165  poimirlem25  33105
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