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Theorem eueq2 3110
 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 117 . . . 4
2 eueq2.1 . . . . . 6
32eueq1 3109 . . . . 5
4 euanv 2344 . . . . . 6
54biimpri 199 . . . . 5
63, 5mpan2 654 . . . 4
7 euorv 2311 . . . 4
81, 6, 7syl2anc 644 . . 3
9 orcom 378 . . . . 5
101bianfd 894 . . . . . 6
1110orbi2d 684 . . . . 5
129, 11syl5bb 250 . . . 4
1312eubidv 2291 . . 3
148, 13mpbid 203 . 2
15 eueq2.2 . . . . . 6
1615eueq1 3109 . . . . 5
17 euanv 2344 . . . . . 6
1817biimpri 199 . . . . 5
1916, 18mpan2 654 . . . 4
20 euorv 2311 . . . 4
2119, 20mpdan 651 . . 3
22 id 21 . . . . . 6
2322bianfd 894 . . . . 5
2423orbi1d 685 . . . 4
2524eubidv 2291 . . 3
2621, 25mpbid 203 . 2
2714, 26pm2.61i 159 1
 Colors of variables: wff set class Syntax hints:   wn 3   wo 359   wa 360   wceq 1653   wcel 1726  weu 2283  cvv 2958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
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