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Theorem reusv3i 4659
Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1
reusv3.2
Assertion
Ref Expression
reusv3i
Distinct variable groups:   , , ,   , ,   , ,   , ,   , ,

Proof of Theorem reusv3i
StepHypRef Expression
1 reusv3.1 . . . . . 6
2 reusv3.2 . . . . . . 7
32eqeq2d 2471 . . . . . 6
41, 3imbi12d 320 . . . . 5
54cbvralv 3084 . . . 4
65biimpi 194 . . 3
7 raaanv 3938 . . . 4
8 prth 571 . . . . . . 7
9 eqtr2 2484 . . . . . . 7
108, 9syl6 33 . . . . . 6
1110ralimi 2850 . . . . 5
1211ralimi 2850 . . . 4
137, 12sylbir 213 . . 3
146, 13mpdan 668 . 2
1514rexlimivw 2946 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  A.wral 2807  E.wrex 2808
This theorem is referenced by:  reusv3  4660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-nul 3785
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