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Theorem reu6 3547
Description: A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
Assertion
Ref Expression
reu6 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reu6
StepHypRef Expression
1 df-reu 3068 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 19.28v 2077 . . . . 5 (∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦))))
3 eleq1w 2833 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4 sbequ12 2267 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
53, 4anbi12d 616 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6 equequ1 2110 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 = 𝑦𝑦 = 𝑦))
75, 6bibi12d 334 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦)))
8 equid 2097 . . . . . . . . . . . 12 𝑦 = 𝑦
98tbt 358 . . . . . . . . . . 11 ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦))
10 simpl 468 . . . . . . . . . . 11 ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) → 𝑦𝐴)
119, 10sylbir 225 . . . . . . . . . 10 (((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦) → 𝑦𝐴)
127, 11syl6bi 243 . . . . . . . . 9 (𝑥 = 𝑦 → (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → 𝑦𝐴))
1312spimv 2419 . . . . . . . 8 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → 𝑦𝐴)
14 ibar 518 . . . . . . . . . . 11 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
1514bibi1d 332 . . . . . . . . . 10 (𝑥𝐴 → ((𝜑𝑥 = 𝑦) ↔ ((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦)))
1615biimprcd 240 . . . . . . . . 9 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1716sps 2209 . . . . . . . 8 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1813, 17jca 501 . . . . . . 7 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
1918axc4i 2295 . . . . . 6 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → ∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
20 biimp 205 . . . . . . . . . . 11 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
2120imim2i 16 . . . . . . . . . 10 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
2221impd 396 . . . . . . . . 9 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → ((𝑥𝐴𝜑) → 𝑥 = 𝑦))
2322adantl 467 . . . . . . . 8 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ((𝑥𝐴𝜑) → 𝑥 = 𝑦))
24 eleq1a 2845 . . . . . . . . . . . 12 (𝑦𝐴 → (𝑥 = 𝑦𝑥𝐴))
2524adantr 466 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → (𝑥 = 𝑦𝑥𝐴))
2625imp 393 . . . . . . . . . 10 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥𝐴)
27 biimpr 210 . . . . . . . . . . . . . 14 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
2827imim2i 16 . . . . . . . . . . . . 13 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥𝐴 → (𝑥 = 𝑦𝜑)))
2928com23 86 . . . . . . . . . . . 12 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
3029imp 393 . . . . . . . . . . 11 (((𝑥𝐴 → (𝜑𝑥 = 𝑦)) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3130adantll 693 . . . . . . . . . 10 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3226, 31jcai 506 . . . . . . . . 9 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3332ex 397 . . . . . . . 8 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
3423, 33impbid 202 . . . . . . 7 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
3534alimi 1887 . . . . . 6 (∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
3619, 35impbii 199 . . . . 5 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
37 df-ral 3066 . . . . . 6 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
3837anbi2i 609 . . . . 5 ((𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦))))
392, 36, 383bitr4i 292 . . . 4 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
4039exbii 1924 . . 3 (∃𝑦𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
41 df-eu 2622 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
42 df-rex 3067 . . 3 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
4340, 41, 423bitr4i 292 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
441, 43bitri 264 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629  wex 1852  [wsb 2049  wcel 2145  ∃!weu 2618  wral 3061  wrex 3062  ∃!wreu 3063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-cleq 2764  df-clel 2767  df-ral 3066  df-rex 3067  df-reu 3068
This theorem is referenced by:  reu3  3548  reu6i  3549  reu8  3554  xpf1o  8278  ufileu  21943  isppw2  25062  cusgrfilem2  26587  fgreu  29811  fcnvgreu  29812  fourierdlem50  40890
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