ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reu7 GIF version

Theorem reu7 2810
Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
Hypothesis
Ref Expression
rmo4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
reu7 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem reu7
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 reu3 2805 . 2 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧)))
2 rmo4.1 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
3 equequ1 1645 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
4 equcom 1639 . . . . . . . 8 (𝑦 = 𝑧𝑧 = 𝑦)
53, 4syl6bb 194 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦))
62, 5imbi12d 232 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜓𝑧 = 𝑦)))
76cbvralv 2590 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∀𝑦𝐴 (𝜓𝑧 = 𝑦))
87rexbii 2385 . . . 4 (∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∃𝑧𝐴𝑦𝐴 (𝜓𝑧 = 𝑦))
9 equequ1 1645 . . . . . . 7 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
109imbi2d 228 . . . . . 6 (𝑧 = 𝑥 → ((𝜓𝑧 = 𝑦) ↔ (𝜓𝑥 = 𝑦)))
1110ralbidv 2380 . . . . 5 (𝑧 = 𝑥 → (∀𝑦𝐴 (𝜓𝑧 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
1211cbvrexv 2591 . . . 4 (∃𝑧𝐴𝑦𝐴 (𝜓𝑧 = 𝑦) ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦))
138, 12bitri 182 . . 3 (∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦))
1413anbi2i 445 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧)) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
151, 14bitri 182 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wral 2359  wrex 2360  ∃!wreu 2361
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator