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Theorem eu6lem 2644
 Description: Proof lines shared by eu6 2645 and eu6im 2646. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2640 was then proved as dfeu 2667. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Extract common proof lines. (Revised by Wolf Lammen, 3-Mar-2023.)
Assertion
Ref Expression
eu6lem (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eu6lem
StepHypRef Expression
1 19.42v 2052 . . . 4 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
2 pm3.33 781 . . . . . . . . 9 (((𝑥 = 𝑦𝜑) ∧ (𝜑𝑥 = 𝑧)) → (𝑥 = 𝑦𝑥 = 𝑧))
32alanimi 1915 . . . . . . . 8 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧))
4 equvelv 2137 . . . . . . . 8 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) ↔ 𝑦 = 𝑧)
53, 4sylib 210 . . . . . . 7 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧)
65pm4.71i 555 . . . . . 6 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ∧ 𝑦 = 𝑧))
7 albiim 1991 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
87biancomi 456 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
9 equequ2 2130 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
109imbi2d 332 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
1110albidv 2019 . . . . . . . . 9 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
1211anbi2d 622 . . . . . . . 8 (𝑦 = 𝑧 → ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧))))
138, 12syl5bb 275 . . . . . . 7 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧))))
1413pm5.32ri 571 . . . . . 6 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧) ↔ ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ∧ 𝑦 = 𝑧))
156, 14bitr4i 270 . . . . 5 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
1615exbii 1947 . . . 4 (∃𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
17 ax6evr 2119 . . . . 5 𝑧 𝑦 = 𝑧
1817biantru 525 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
191, 16, 183bitr4ri 296 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)))
2019exbii 1947 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑦𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)))
21 exdistrv 2054 . 2 (∃𝑦𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
2220, 21bitri 267 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654  ∃wex 1878 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879 This theorem is referenced by:  eu6  2645  eu6im  2646
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