ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reuun1 GIF version

Theorem reuun1 3281
Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
Assertion
Ref Expression
reuun1 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reuun1
StepHypRef Expression
1 ssun1 3163 . 2 𝐴 ⊆ (𝐴𝐵)
2 orc 668 . . 3 (𝜑 → (𝜑𝜓))
32rgenw 2430 . 2 𝑥𝐴 (𝜑 → (𝜑𝜓))
4 reuss2 3279 . 2 (((𝐴 ⊆ (𝐴𝐵) ∧ ∀𝑥𝐴 (𝜑 → (𝜑𝜓))) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓))) → ∃!𝑥𝐴 𝜑)
51, 3, 4mpanl12 427 1 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 664  wral 2359  wrex 2360  ∃!wreu 2361  cun 2997  wss 2999
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-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-un 3003  df-in 3005  df-ss 3012
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator