ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reu6 GIF version

Theorem reu6 2792
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 2360 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 19.28v 1823 . . . . 5 (∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦))))
3 eleq1 2145 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4 sbequ12 1696 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
53, 4anbi12d 457 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6 equequ1 1640 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 = 𝑦𝑦 = 𝑦))
75, 6bibi12d 233 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦)))
8 equid 1630 . . . . . . . . . . . 12 𝑦 = 𝑦
98tbt 245 . . . . . . . . . . 11 ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦))
10 simpl 107 . . . . . . . . . . 11 ((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) → 𝑦𝐴)
119, 10sylbir 133 . . . . . . . . . 10 (((𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦) → 𝑦𝐴)
127, 11syl6bi 161 . . . . . . . . 9 (𝑥 = 𝑦 → (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → 𝑦𝐴))
1312spimv 1734 . . . . . . . 8 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → 𝑦𝐴)
14 bi1 116 . . . . . . . . . . . 12 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → ((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1514expdimp 255 . . . . . . . . . . 11 ((((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ∧ 𝑥𝐴) → (𝜑𝑥 = 𝑦))
16 bi2 128 . . . . . . . . . . . . 13 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
17 simpr 108 . . . . . . . . . . . . 13 ((𝑥𝐴𝜑) → 𝜑)
1816, 17syl6 33 . . . . . . . . . . . 12 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
1918adantr 270 . . . . . . . . . . 11 ((((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ∧ 𝑥𝐴) → (𝑥 = 𝑦𝜑))
2015, 19impbid 127 . . . . . . . . . 10 ((((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ∧ 𝑥𝐴) → (𝜑𝑥 = 𝑦))
2120ex 113 . . . . . . . . 9 (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
2221sps 1471 . . . . . . . 8 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
2313, 22jca 300 . . . . . . 7 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → (𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
2423a5i 1476 . . . . . 6 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) → ∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
25 bi1 116 . . . . . . . . . . 11 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
2625imim2i 12 . . . . . . . . . 10 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
2726impd 251 . . . . . . . . 9 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → ((𝑥𝐴𝜑) → 𝑥 = 𝑦))
2827adantl 271 . . . . . . . 8 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ((𝑥𝐴𝜑) → 𝑥 = 𝑦))
29 eleq1a 2154 . . . . . . . . . . . 12 (𝑦𝐴 → (𝑥 = 𝑦𝑥𝐴))
3029adantr 270 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → (𝑥 = 𝑦𝑥𝐴))
3130imp 122 . . . . . . . . . 10 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥𝐴)
32 bi2 128 . . . . . . . . . . . . . 14 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
3332imim2i 12 . . . . . . . . . . . . 13 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥𝐴 → (𝑥 = 𝑦𝜑)))
3433com23 77 . . . . . . . . . . . 12 ((𝑥𝐴 → (𝜑𝑥 = 𝑦)) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
3534imp 122 . . . . . . . . . . 11 (((𝑥𝐴 → (𝜑𝑥 = 𝑦)) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3635adantll 460 . . . . . . . . . 10 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3731, 36jcai 304 . . . . . . . . 9 (((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥𝐴𝜑))
3837ex 113 . . . . . . . 8 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
3928, 38impbid 127 . . . . . . 7 ((𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
4039alimi 1385 . . . . . 6 (∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))) → ∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
4124, 40impbii 124 . . . . 5 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑦𝐴 ∧ (𝑥𝐴 → (𝜑𝑥 = 𝑦))))
42 df-ral 2358 . . . . . 6 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
4342anbi2i 445 . . . . 5 ((𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦))))
442, 41, 433bitr4i 210 . . . 4 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
4544exbii 1537 . . 3 (∃𝑦𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
46 df-eu 1946 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑦))
47 df-rex 2359 . . 3 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝑦)))
4845, 46, 473bitr4i 210 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
491, 48bitri 182 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wex 1422  wcel 1434  [wsb 1687  ∃!weu 1943  wral 2353  wrex 2354  ∃!wreu 2355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-cleq 2076  df-clel 2079  df-ral 2358  df-rex 2359  df-reu 2360
This theorem is referenced by:  reu3  2793  reu6i  2794  reu8  2799  xpf1o  6490
  Copyright terms: Public domain W3C validator