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Theorem reu3 3382
Description: A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
Assertion
Ref Expression
reu3 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reu3
StepHypRef Expression
1 reurex 3152 . . 3 (∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)
2 reu6 3381 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
3 biimp 205 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
43ralimi 2947 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) → ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
54reximi 3006 . . . 4 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
62, 5sylbi 207 . . 3 (∃!𝑥𝐴 𝜑 → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
71, 6jca 554 . 2 (∃!𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
8 rexex 2997 . . . 4 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
98anim2i 592 . . 3 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)) → (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)))
10 eu3v 2497 . . . 4 (∃!𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦)))
11 df-reu 2914 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
12 df-rex 2913 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
13 df-ral 2912 . . . . . . 7 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
14 impexp 462 . . . . . . . 8 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1514albii 1744 . . . . . . 7 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1613, 15bitr4i 267 . . . . . 6 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1716exbii 1771 . . . . 5 (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1812, 17anbi12i 732 . . . 4 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦)))
1910, 11, 183bitr4i 292 . . 3 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)))
209, 19sylibr 224 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)) → ∃!𝑥𝐴 𝜑)
217, 20impbii 199 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701  wcel 1987  ∃!weu 2469  wral 2907  wrex 2908  ∃!wreu 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-cleq 2614  df-clel 2617  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915
This theorem is referenced by:  reu7  3387  2reu4a  40519
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