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Theorem cbvreu 3354
Description: Change the bound variable of a restricted unique existential 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 2005 . . . 4 𝑧(𝑥𝐴𝜑)
21sb8eu 2663 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑))
3 sban 2557 . . . 4 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
43eubii 2651 . . 3 (∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
5 clelsb3 2909 . . . . . 6 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
65anbi1i 612 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
76eubii 2651 . . . 4 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
8 nfv 2005 . . . . . 6 𝑦 𝑧𝐴
9 cbvral.1 . . . . . . 7 𝑦𝜑
109nfsb 2601 . . . . . 6 𝑦[𝑧 / 𝑥]𝜑
118, 10nfan 1990 . . . . 5 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
12 nfv 2005 . . . . 5 𝑧(𝑦𝐴𝜓)
13 eleq1w 2864 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
14 sbequ 2534 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 cbvral.2 . . . . . . . 8 𝑥𝜓
16 cbvral.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16sbie 2566 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜓)
1814, 17syl6bb 278 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
1913, 18anbi12d 618 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2011, 12, 19cbveu 2665 . . . 4 (∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
217, 20bitri 266 . . 3 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
222, 4, 213bitri 288 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
23 df-reu 3099 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
24 df-reu 3099 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2522, 23, 243bitr4i 294 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wnf 1863  [wsb 2059  wcel 2155  ∃!weu 2629  ∃!wreu 3094
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 2067  ax-7 2103  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419
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 2060  df-eu 2633  df-clel 2798  df-reu 3099
This theorem is referenced by:  cbvrmo  3355  cbvreuv  3358  reu8nf  3705  poimirlem25  33741
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