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Theorem eu6lem 2657
Description: Lemma of eu6im 2659. A dissection of an idiom characterizing existential uniqueness. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2653 was then proved as dfeu 2680. (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 1954 . . . 4 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
2 alsyl 1894 . . . . . . . 8 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧))
3 equvelv 2038 . . . . . . . 8 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) ↔ 𝑦 = 𝑧)
42, 3sylib 220 . . . . . . 7 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧)
54pm4.71i 562 . . . . . 6 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ∧ 𝑦 = 𝑧))
6 albiim 1890 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
76biancomi 465 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
8 equequ2 2033 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
98imbi2d 343 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
109albidv 1921 . . . . . . . . 9 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
1110anbi2d 630 . . . . . . . 8 (𝑦 = 𝑧 → ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧))))
127, 11syl5bb 285 . . . . . . 7 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧))))
1312pm5.32ri 578 . . . . . 6 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧) ↔ ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ∧ 𝑦 = 𝑧))
145, 13bitr4i 280 . . . . 5 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
1514exbii 1848 . . . 4 (∃𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
16 ax6evr 2022 . . . . 5 𝑧 𝑦 = 𝑧
1716biantru 532 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
181, 15, 173bitr4ri 306 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)))
1918exbii 1848 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑦𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)))
20 exdistrv 1956 . 2 (∃𝑦𝑧(∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
2119, 20bitri 277 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1535  wex 1780
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 1911  ax-6 1970  ax-7 2015
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781
This theorem is referenced by:  eu6im  2659
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