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Theorem reuun1 3353
 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 3234 . 2 𝐴 ⊆ (𝐴𝐵)
2 orc 701 . . 3 (𝜑 → (𝜑𝜓))
32rgenw 2485 . 2 𝑥𝐴 (𝜑 → (𝜑𝜓))
4 reuss2 3351 . 2 (((𝐴 ⊆ (𝐴𝐵) ∧ ∀𝑥𝐴 (𝜑 → (𝜑𝜓))) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓))) → ∃!𝑥𝐴 𝜑)
51, 3, 4mpanl12 432 1 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 697  ∀wral 2414  ∃wrex 2415  ∃!wreu 2416   ∪ cun 3064   ⊆ wss 3066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-v 2683  df-un 3070  df-in 3072  df-ss 3079 This theorem is referenced by: (None)
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