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Theorem reu3 3736
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 3382 . . 3 (∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)
2 reu6 3735 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
3 biimp 215 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
43ralimi 3081 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) → ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
54reximi 3082 . . . 4 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
62, 5sylbi 217 . . 3 (∃!𝑥𝐴 𝜑 → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
71, 6jca 511 . 2 (∃!𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
8 rexex 3074 . . . 4 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
98anim2i 617 . . 3 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)) → (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)))
10 eu3v 2568 . . . 4 (∃!𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦)))
11 df-reu 3379 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
12 df-rex 3069 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
13 df-ral 3060 . . . . . . 7 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
14 impexp 450 . . . . . . . 8 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1514albii 1816 . . . . . . 7 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1613, 15bitr4i 278 . . . . . 6 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1716exbii 1845 . . . . 5 (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1812, 17anbi12i 628 . . . 4 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦)))
1910, 11, 183bitr4i 303 . . 3 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)))
209, 19sylibr 234 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)) → ∃!𝑥𝐴 𝜑)
217, 20impbii 209 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wex 1776  wcel 2106  ∃!weu 2566  wral 3059  wrex 3068  ∃!wreu 3376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clel 2814  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379
This theorem is referenced by:  reu7  3741  2reu4lem  4528  reu3op  6314
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