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Theorem reuun1 3213
Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
Assertion
Ref Expression
reuun1 ((x A φ ∃!x (AB)(φ ψ)) → ∃!x A φ)
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem reuun1
StepHypRef Expression
1 ssun1 3100 . 2 A ⊆ (AB)
2 orc 632 . . 3 (φ → (φ ψ))
32rgenw 2370 . 2 x A (φ → (φ ψ))
4 reuss2 3211 . 2 (((A ⊆ (AB) x A (φ → (φ ψ))) (x A φ ∃!x (AB)(φ ψ))) → ∃!x A φ)
51, 3, 4mpanl12 412 1 ((x A φ ∃!x (AB)(φ ψ)) → ∃!x A φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628  wral 2300  wrex 2301  ∃!wreu 2302  cun 2909  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-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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