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Theorem reuss 3280
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 idd 21 . . . 4 (𝑥𝐴 → (𝜑𝜑))
21rgen 2428 . . 3 𝑥𝐴 (𝜑𝜑)
3 reuss2 3279 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜑)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ∃!𝑥𝐴 𝜑)
42, 3mpanl2 426 . 2 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ∃!𝑥𝐴 𝜑)
543impb 1139 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924  wcel 1438  wral 2359  wrex 2360  ∃!wreu 2361  wss 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-rex 2365  df-reu 2366  df-in 3005  df-ss 3012
This theorem is referenced by:  riotass  5635
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