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Theorem reuss 3941
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 2953 . . 3 𝑥𝐴 (𝜑𝜑)
3 reuss2 3940 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜑)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ∃!𝑥𝐴 𝜑)
42, 3mpanl2 717 . 2 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ∃!𝑥𝐴 𝜑)
543impb 1279 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wral 2941  wrex 2942  ∃!wreu 2943  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946  df-rex 2947  df-reu 2948  df-in 3614  df-ss 3621
This theorem is referenced by:  euelss  3947  riotass  6679  adjbdln  29070
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