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Theorem reusv3i 4157
Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1
reusv3.2  C  D
Assertion
Ref Expression
reusv3i  C  C  D
Distinct variable groups:   ,,,   , C,   , D,   ,,   ,,
Allowed substitution hints:   ()   ()   (,,)    C()    D()

Proof of Theorem reusv3i
StepHypRef Expression
1 reusv3.1 . . . . . 6
2 reusv3.2 . . . . . . 7  C  D
32eqeq2d 2048 . . . . . 6  C  D
41, 3imbi12d 223 . . . . 5  C  D
54cbvralv 2527 . . . 4  C  D
65biimpi 113 . . 3  C  D
7 raaanv 3322 . . . 4  C  D  C  D
8 prth 326 . . . . . . 7  C  D  C  D
9 eqtr2 2055 . . . . . . 7  C  D  C  D
108, 9syl6 29 . . . . . 6  C  D  C  D
1110ralimi 2378 . . . . 5  C  D  C  D
1211ralimi 2378 . . . 4  C  D  C  D
137, 12sylbir 125 . . 3  C  D  C  D
146, 13mpdan 398 . 2  C  C  D
1514rexlimivw 2423 1  C  C  D
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wral 2300  wrex 2301
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-bnd 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-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306
This theorem is referenced by:  reusv3  4158
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