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Theorem reuss 3536
 Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reuss ((A B x A φ ∃!x B φ) → ∃!x A φ)
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem reuss
StepHypRef Expression
1 idd 21 . . . 4 (x A → (φφ))
21rgen 2679 . . 3 x A (φφ)
3 reuss2 3535 . . 3 (((A B x A (φφ)) (x A φ ∃!x B φ)) → ∃!x A φ)
42, 3mpanl2 662 . 2 ((A B (x A φ ∃!x B φ)) → ∃!x A φ)
543impb 1147 1 ((A B x A φ ∃!x B φ) → ∃!x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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