MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reuun1 Structured version   Visualization version   GIF version

Theorem reuun1 4288
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 4151 . 2 𝐴 ⊆ (𝐴𝐵)
2 orc 863 . . 3 (𝜑 → (𝜑𝜓))
32rgenw 3154 . 2 𝑥𝐴 (𝜑 → (𝜑𝜓))
4 reuss2 4286 . 2 (((𝐴 ⊆ (𝐴𝐵) ∧ ∀𝑥𝐴 (𝜑 → (𝜑𝜓))) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓))) → ∃!𝑥𝐴 𝜑)
51, 3, 4mpanl12 698 1 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 843  wral 3142  wrex 3143  ∃!wreu 3144  cun 3937  wss 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-reu 3149  df-v 3501  df-un 3944  df-in 3946  df-ss 3955
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator