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Theorem ceqex 2665
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 1479 . . 3
2 isset 2555 . . 3  _V
31, 2sylibr 137 . 2  _V
4 eqeq2 2046 . . . 4
54anbi1d 438 . . . . . 6
65exbidv 1703 . . . . 5
76bibi2d 221 . . . 4
84, 7imbi12d 223 . . 3
9 19.8a 1479 . . . . 5
109ex 108 . . . 4
11 vex 2554 . . . . . 6 
_V
1211alexeq 2664 . . . . 5
13 sp 1398 . . . . . 6
1413com12 27 . . . . 5
1512, 14syl5bir 142 . . . 4
1610, 15impbid 120 . . 3
178, 16vtoclg 2607 . 2  _V
183, 17mpcom 32 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242  wex 1378   wcel 1390   _Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  ceqsexg  2666  sbc6g  2782
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