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Theorem reuss 4333
Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reuss ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reuss
StepHypRef Expression
1 id 22 . . . 4 (𝜑𝜑)
21rgenw 3063 . . 3 𝑥𝐴 (𝜑𝜑)
3 reuss2 4332 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜑)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ∃!𝑥𝐴 𝜑)
42, 3mpanl2 701 . 2 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ∃!𝑥𝐴 𝜑)
543impb 1114 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wral 3059  wrex 3068  ∃!wreu 3376  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1777  df-mo 2538  df-eu 2567  df-clel 2814  df-ral 3060  df-rex 3069  df-reu 3379  df-ss 3980
This theorem is referenced by:  euelss  4338  riotass  7419  adjbdln  32112  tfsconcatlem  43326  tfsconcatfv  43331
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