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Theorem reusv3i 4722
 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 2446 . . . . . 6
41, 3imbi12d 312 . . . . 5
54cbvralv 2924 . . . 4
65biimpi 187 . . 3
7 raaanv 3728 . . . 4
8 prth 555 . . . . . . 7
9 eqtr2 2453 . . . . . . 7
108, 9syl6 31 . . . . . 6
1110ralimi 2773 . . . . 5
1211ralimi 2773 . . . 4
137, 12sylbir 205 . . 3
146, 13mpdan 650 . 2
1514rexlimivw 2818 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652  wral 2697  wrex 2698 This theorem is referenced by:  reusv3  4723 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-nul 3621
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