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Theorem cbvreuwOLD 3411
Description: Obsolete version of cbvreuw 3405 as of 10-Dec-2024. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cbvreuwOLD.1 𝑦𝜑
cbvreuwOLD.2 𝑥𝜓
cbvreuwOLD.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreuwOLD (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvreuwOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1916 . . . 4 𝑧(𝑥𝐴𝜑)
21sb8euv 2592 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑))
3 sban 2082 . . . 4 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
43eubii 2578 . . 3 (∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
5 clelsb1 2859 . . . . . 6 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
65anbi1i 623 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
76eubii 2578 . . . 4 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
8 nfv 1916 . . . . . 6 𝑦 𝑧𝐴
9 cbvreuwOLD.1 . . . . . . 7 𝑦𝜑
109nfsbv 2322 . . . . . 6 𝑦[𝑧 / 𝑥]𝜑
118, 10nfan 1901 . . . . 5 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
12 nfv 1916 . . . . 5 𝑧(𝑦𝐴𝜓)
13 eleq1w 2815 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
14 sbequ 2085 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 cbvreuwOLD.2 . . . . . . . 8 𝑥𝜓
16 cbvreuwOLD.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16sbiev 2307 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜓)
1814, 17bitrdi 286 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
1913, 18anbi12d 630 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2011, 12, 19cbveuw 2600 . . . 4 (∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
217, 20bitri 274 . . 3 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
222, 4, 213bitri 296 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
23 df-reu 3376 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
24 df-reu 3376 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2522, 23, 243bitr4i 302 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wnf 1784  [wsb 2066  wcel 2105  ∃!weu 2561  ∃!wreu 3373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-10 2136  ax-11 2153  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clel 2809  df-reu 3376
This theorem is referenced by: (None)
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