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Theorem eu6OLD 2594
 Description: Obsolete version of eu6 2592 as of 28-Dec-2022. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2587 was then proved as dfeu 2614. (Revised by BJ, 30-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eu6OLD (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eu6OLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-eu 2587 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2 exsb 2327 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
3 df-mo 2551 . . 3 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
42, 3anbi12i 620 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
5 exdistrv 1998 . . . 4 (∃𝑦𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
6 19.26 1916 . . . . . . . 8 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)))
7 pm3.33 755 . . . . . . . . . . 11 (((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) → (𝑥 = 𝑦𝑥 = 𝑧))
87pm4.71i 555 . . . . . . . . . 10 (((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ↔ (((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ∧ (𝑥 = 𝑦𝑥 = 𝑧)))
98albii 1863 . . . . . . . . 9 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ↔ ∀𝑥(((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ∧ (𝑥 = 𝑦𝑥 = 𝑧)))
10 19.26 1916 . . . . . . . . 9 (∀𝑥(((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ∧ (𝑥 = 𝑦𝑥 = 𝑧)) ↔ (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ∧ ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
11 equvelv 2080 . . . . . . . . . . 11 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) ↔ 𝑦 = 𝑧)
126, 11anbi12i 620 . . . . . . . . . 10 ((∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ∧ ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧)) ↔ ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ∧ 𝑦 = 𝑧))
13 anass 462 . . . . . . . . . . 11 (((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ (∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑦 = 𝑧)))
14 equequ2 2073 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
1514bicomd 215 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑥 = 𝑧𝑥 = 𝑦))
1615imbi2d 332 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
1716albidv 1963 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1817pm5.32ri 571 . . . . . . . . . . . 12 ((∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
1918anbi2i 616 . . . . . . . . . . 11 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ (∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑦 = 𝑧)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧)))
2013, 19bitri 267 . . . . . . . . . 10 (((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧)))
21 anass 462 . . . . . . . . . . 11 (((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧)))
22 19.26 1916 . . . . . . . . . . . . 13 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑦)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
23 ancom 454 . . . . . . . . . . . . . . 15 (((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑦)) ↔ ((𝜑𝑥 = 𝑦) ∧ (𝑥 = 𝑦𝜑)))
24 dfbi2 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 = 𝑦) ↔ ((𝜑𝑥 = 𝑦) ∧ (𝑥 = 𝑦𝜑)))
2524bicomi 216 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = 𝑦) ∧ (𝑥 = 𝑦𝜑)) ↔ (𝜑𝑥 = 𝑦))
2623, 25bitri 267 . . . . . . . . . . . . . 14 (((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑦)) ↔ (𝜑𝑥 = 𝑦))
2726albii 1863 . . . . . . . . . . . . 13 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑦)) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
2822, 27bitr3i 269 . . . . . . . . . . . 12 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
2928anbi1i 617 . . . . . . . . . . 11 (((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
3021, 29bitr3i 269 . . . . . . . . . 10 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
3112, 20, 303bitri 289 . . . . . . . . 9 ((∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ∧ ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
329, 10, 313bitri 289 . . . . . . . 8 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
336, 32bitr3i 269 . . . . . . 7 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
3433exbii 1892 . . . . . 6 (∃𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
35 19.42v 1996 . . . . . 6 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
3634, 35bitri 267 . . . . 5 (∃𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
3736exbii 1892 . . . 4 (∃𝑦𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
385, 37bitr3i 269 . . 3 ((∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)) ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
39 ax6evr 2062 . . . . . 6 𝑧 𝑦 = 𝑧
4039biantru 525 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
4140bicomi 216 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
4241exbii 1892 . . 3 (∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
4338, 42bitri 267 . 2 ((∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
441, 4, 433bitri 289 1 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599  ∃wex 1823  ∃*wmo 2549  ∃!weu 2586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-mo 2551  df-eu 2587 This theorem is referenced by: (None)
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