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Theorem ceqsalg 2576
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
ceqsalg.1  F/
ceqsalg.2
Assertion
Ref Expression
ceqsalg  V
Distinct variable group:   ,
Allowed substitution hints:   ()   ()    V()

Proof of Theorem ceqsalg
StepHypRef Expression
1 elisset 2562 . . 3  V
2 nfa1 1431 . . . 4  F/
3 ceqsalg.1 . . . 4  F/
4 ceqsalg.2 . . . . . . 7
54biimpd 132 . . . . . 6
65a2i 11 . . . . 5
76sps 1427 . . . 4
82, 3, 7exlimd 1485 . . 3
91, 8syl5com 26 . 2  V
104biimprcd 149 . . 3
113, 10alrimi 1412 . 2
129, 11impbid1 130 1  V
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98  wal 1240   wceq 1242   F/wnf 1346  wex 1378   wcel 1390
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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  ceqsal  2577  sbc6g  2782  uniiunlem  3022  sucprcreg  4227  funimass4  5167  ralrnmpt2  5557
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