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Theorem reusv3i 4380
 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 2151 . . . . . 6
41, 3imbi12d 233 . . . . 5
54cbvralv 2654 . . . 4
65biimpi 119 . . 3
7 raaanv 3470 . . . 4
8 anim12 341 . . . . . . 7
9 eqtr2 2158 . . . . . . 7
108, 9syl6 33 . . . . . 6
1110ralimi 2495 . . . . 5
1211ralimi 2495 . . . 4
137, 12sylbir 134 . . 3
146, 13mpdan 417 . 2
1514rexlimivw 2545 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1331  wral 2416  wrex 2417 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422 This theorem is referenced by:  reusv3  4381
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