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Theorem reuss 3212
 Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reuss ((AB 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 2368 . . 3 x A (φφ)
3 reuss2 3211 . . 3 (((AB x A (φφ)) (x A φ ∃!x B φ)) → ∃!x A φ)
42, 3mpanl2 411 . 2 ((AB (x A φ ∃!x B φ)) → ∃!x A φ)
543impb 1099 1 ((AB x A φ ∃!x B φ) → ∃!x A φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  ∃!wreu 2302   ⊆ wss 2911 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307  df-in 2918  df-ss 2925 This theorem is referenced by:  riotass  5438
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