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Theorem reusv3i 4191
 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:   ,,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   ()   ()   (,,)   ()   ()

Proof of Theorem reusv3i
StepHypRef Expression
1 reusv3.1 . . . . . 6
2 reusv3.2 . . . . . . 7
32eqeq2d 2051 . . . . . 6
41, 3imbi12d 223 . . . . 5
54cbvralv 2533 . . . 4
65biimpi 113 . . 3
7 raaanv 3328 . . . 4
8 prth 326 . . . . . . 7
9 eqtr2 2058 . . . . . . 7
108, 9syl6 29 . . . . . 6
1110ralimi 2384 . . . . 5
1211ralimi 2384 . . . 4
137, 12sylbir 125 . . 3
146, 13mpdan 398 . 2
1514rexlimivw 2429 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wral 2306  wrex 2307 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312 This theorem is referenced by:  reusv3  4192
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