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Theorem reu6 2941
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 2475 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 19.28v 1912 . . . . 5 (∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦))))
3 eleq1 2252 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4 sbequ12 1782 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
53, 4anbi12d 473 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6 equequ1 1723 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 = 𝑦𝑦 = 𝑦))
75, 6bibi12d 235 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦)))
8 equid 1712 . . . . . . . . . . . 12 𝑦 = 𝑦
98tbt 247 . . . . . . . . . . 11 ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦))
10 simpl 109 . . . . . . . . . . 11 ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) → 𝑦𝐴)
119, 10sylbir 135 . . . . . . . . . 10 (((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦) → 𝑦𝐴)
127, 11biimtrdi 163 . . . . . . . . 9 (𝑥 = 𝑦 → (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → 𝑦𝐴))
1312spimv 1822 . . . . . . . 8 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → 𝑦𝐴)
14 biimp 118 . . . . . . . . . . . 12 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → ((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1514expdimp 259 . . . . . . . . . . 11 ((((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ∧ 𝑥𝐴) → (𝜑𝑥 = 𝑦))
16 biimpr 130 . . . . . . . . . . . . 13 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
17 simpr 110 . . . . . . . . . . . . 13 ((𝑥𝐴𝜑) → 𝜑)
1816, 17syl6 33 . . . . . . . . . . . 12 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
1918adantr 276 . . . . . . . . . . 11 ((((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ∧ 𝑥𝐴) → (𝑥 = 𝑦𝜑))
2015, 19impbid 129 . . . . . . . . . 10 ((((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ∧ 𝑥𝐴) → (𝜑𝑥 = 𝑦))
2120ex 115 . . . . . . . . 9 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
2221sps 1548 . . . . . . . 8 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
2313, 22jca 306 . . . . . . 7 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
2423a5i 1554 . . . . . 6 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → ∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
25 biimp 118 . . . . . . . . . . 11 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
2625imim2i 12 . . . . . . . . . 10 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
2726impd 254 . . . . . . . . 9 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → ((𝑥𝐴𝜑) → 𝑥 = 𝑦))
2827adantl 277 . . . . . . . 8 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ((𝑥𝐴𝜑) → 𝑥 = 𝑦))
29 eleq1a 2261 . . . . . . . . . . . 12 (𝑦𝐴 → (𝑥 = 𝑦𝑥𝐴))
3029adantr 276 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → (𝑥 = 𝑦𝑥𝐴))
3130imp 124 . . . . . . . . . 10 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥𝐴)
32 biimpr 130 . . . . . . . . . . . . . 14 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
3332imim2i 12 . . . . . . . . . . . . 13 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥𝐴 → (𝑥 = 𝑦𝜑)))
3433com23 78 . . . . . . . . . . . 12 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
3534imp 124 . . . . . . . . . . 11 (((𝑥𝐴 → (𝜑𝑥 = 𝑦)) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3635adantll 476 . . . . . . . . . 10 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3731, 36jcai 311 . . . . . . . . 9 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3837ex 115 . . . . . . . 8 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
3928, 38impbid 129 . . . . . . 7 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
4039alimi 1466 . . . . . 6 (∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
4124, 40impbii 126 . . . . 5 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
42 df-ral 2473 . . . . . 6 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
4342anbi2i 457 . . . . 5 ((𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦))))
442, 41, 433bitr4i 212 . . . 4 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
4544exbii 1616 . . 3 (∃𝑦𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
46 df-eu 2041 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
47 df-rex 2474 . . 3 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
4845, 46, 473bitr4i 212 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
491, 48bitri 184 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362  wex 1503  [wsb 1773  ∃!weu 2038  wcel 2160  wral 2468  wrex 2469  ∃!wreu 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2041  df-cleq 2182  df-clel 2185  df-ral 2473  df-rex 2474  df-reu 2475
This theorem is referenced by:  reu3  2942  reu6i  2943  reu8  2948  xpf1o  6867
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