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Theorem ceqex 2812
 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 1569 . . 3
2 isset 2692 . . 3
31, 2sylibr 133 . 2
4 eqeq2 2149 . . . 4
54anbi1d 460 . . . . . 6
65exbidv 1797 . . . . 5
76bibi2d 231 . . . 4
84, 7imbi12d 233 . . 3
9 19.8a 1569 . . . . 5
109ex 114 . . . 4
11 vex 2689 . . . . . 6
1211alexeq 2811 . . . . 5
13 sp 1488 . . . . . 6
1413com12 30 . . . . 5
1512, 14syl5bir 152 . . . 4
1610, 15impbid 128 . . 3
178, 16vtoclg 2746 . 2
183, 17mpcom 36 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104  wal 1329   wceq 1331  wex 1468   wcel 1480  cvv 2686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688 This theorem is referenced by:  ceqsexg  2813  sbc6g  2933
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