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Theorem reuun1 3246
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 3133 . 2 𝐴 ⊆ (𝐴𝐵)
2 orc 643 . . 3 (𝜑 → (𝜑𝜓))
32rgenw 2393 . 2 𝑥𝐴 (𝜑 → (𝜑𝜓))
4 reuss2 3244 . 2 (((𝐴 ⊆ (𝐴𝐵) ∧ ∀𝑥𝐴 (𝜑 → (𝜑𝜓))) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓))) → ∃!𝑥𝐴 𝜑)
51, 3, 4mpanl12 420 1 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 639  wral 2323  wrex 2324  ∃!wreu 2325  cun 2942  wss 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576  df-un 2949  df-in 2951  df-ss 2958
This theorem is referenced by: (None)
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