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Theorem ceqex 3075
 Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
Assertion
Ref Expression
ceqex
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem ceqex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 19.8a 1765 . . 3
2 isset 2969 . . 3
31, 2sylibr 205 . 2
4 eqeq2 2452 . . . 4
54anbi1d 687 . . . . . 6
65exbidv 1638 . . . . 5
76bibi2d 311 . . . 4
84, 7imbi12d 313 . . 3
9 19.8a 1765 . . . . 5
109ex 425 . . . 4
11 vex 2968 . . . . . 6
1211alexeq 3074 . . . . 5
13 sp 1766 . . . . . 6
1413com12 30 . . . . 5
1512, 14syl5bir 211 . . . 4
1610, 15impbid 185 . . 3
178, 16vtoclg 3020 . 2
183, 17mpcom 35 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wal 1550  wex 1551   wceq 1654   wcel 1728  cvv 2965 This theorem is referenced by:  ceqsexg  3076  sbc6g  3195 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 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424 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 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2967
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