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Theorem reusv3i 4600
 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 2416 . . . . . 6
41, 3imbi12d 318 . . . . 5
54cbvralv 3033 . . . 4
65biimpi 194 . . 3
7 raaanv 3881 . . . 4
8 prth 569 . . . . . . 7
9 eqtr2 2429 . . . . . . 7
108, 9syl6 31 . . . . . 6
1110ralimi 2796 . . . . 5
1211ralimi 2796 . . . 4
137, 12sylbir 213 . . 3
146, 13mpdan 666 . 2
1514rexlimivw 2892 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405  wral 2753  wrex 2754 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-v 3060  df-dif 3416  df-nul 3738 This theorem is referenced by:  reusv3  4601
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