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Theorem eueq2 3010
 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 114 . . . 4
2 eueq2.1 . . . . . 6
32eueq1 3009 . . . . 5
4 euanv 2265 . . . . . 6
54biimpri 197 . . . . 5
63, 5mpan2 652 . . . 4
7 euorv 2232 . . . 4
81, 6, 7syl2anc 642 . . 3
9 orcom 376 . . . . 5
101bianfd 892 . . . . . 6
1110orbi2d 682 . . . . 5
129, 11syl5bb 248 . . . 4
1312eubidv 2212 . . 3
148, 13mpbid 201 . 2
15 eueq2.2 . . . . . 6
1615eueq1 3009 . . . . 5
17 euanv 2265 . . . . . 6
1817biimpri 197 . . . . 5
1916, 18mpan2 652 . . . 4
20 euorv 2232 . . . 4
2119, 20mpdan 649 . . 3
22 id 19 . . . . . 6
2322bianfd 892 . . . . 5
2423orbi1d 683 . . . 4
2524eubidv 2212 . . 3
2621, 25mpbid 201 . 2
2714, 26pm2.61i 156 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wo 357   wa 358   wceq 1642   wcel 1710  weu 2204  cvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by: (None)
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