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Theorem reuun1 3868
 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 3738 . 2 𝐴 ⊆ (𝐴𝐵)
2 orc 399 . . 3 (𝜑 → (𝜑𝜓))
32rgenw 2908 . 2 𝑥𝐴 (𝜑 → (𝜑𝜓))
4 reuss2 3866 . 2 (((𝐴 ⊆ (𝐴𝐵) ∧ ∀𝑥𝐴 (𝜑 → (𝜑𝜓))) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓))) → ∃!𝑥𝐴 𝜑)
51, 3, 4mpanl12 714 1 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∪ cun 3538   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-v 3175  df-un 3545  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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