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Theorem eueq2 3273
 Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
Hypotheses
Ref Expression
eueq2.1
eueq2.2
Assertion
Ref Expression
eueq2
Distinct variable groups:   ,   ,   ,

Proof of Theorem eueq2
StepHypRef Expression
1 notnot1 122 . . . 4
2 eueq2.1 . . . . . 6
32eueq1 3272 . . . . 5
4 euanv 2355 . . . . . 6
54biimpri 206 . . . . 5
63, 5mpan2 671 . . . 4
7 euorv 2333 . . . 4
81, 6, 7syl2anc 661 . . 3
9 orcom 387 . . . . 5
101bianfd 926 . . . . . 6
1110orbi2d 701 . . . . 5
129, 11syl5bb 257 . . . 4
1312eubidv 2305 . . 3
148, 13mpbid 210 . 2
15 eueq2.2 . . . . . 6
1615eueq1 3272 . . . . 5
17 euanv 2355 . . . . . 6
1817biimpri 206 . . . . 5
1916, 18mpan2 671 . . . 4
20 euorv 2333 . . . 4
2119, 20mpdan 668 . . 3
22 id 22 . . . . . 6
2322bianfd 926 . . . . 5
2423orbi1d 702 . . . 4
2524eubidv 2305 . . 3
2621, 25mpbid 210 . 2
2714, 26pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wo 368   wa 369   wceq 1395   wcel 1819  weu 2283  cvv 3109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111 This theorem is referenced by: (None)
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