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Theorem reuss 4287
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 3154 . . 3 𝑥𝐴 (𝜑𝜑)
3 reuss2 4286 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜑)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ∃!𝑥𝐴 𝜑)
42, 3mpanl2 697 . 2 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ∃!𝑥𝐴 𝜑)
543impb 1109 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081  wral 3142  wrex 3143  ∃!wreu 3144  wss 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-ral 3147  df-rex 3148  df-reu 3149  df-in 3946  df-ss 3955
This theorem is referenced by:  euelss  4293  riotass  7140  adjbdln  29774
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