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Theorem gencbvex 2594
Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
gencbvex.1  _V
gencbvex.2
gencbvex.3
gencbvex.4
Assertion
Ref Expression
gencbvex
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()   ()

Proof of Theorem gencbvex
StepHypRef Expression
1 excom 1551 . 2
2 gencbvex.1 . . . 4  _V
3 gencbvex.3 . . . . . . 7
4 gencbvex.2 . . . . . . 7
53, 4anbi12d 442 . . . . . 6
65bicomd 129 . . . . 5
76eqcoms 2040 . . . 4
82, 7ceqsexv 2587 . . 3
98exbii 1493 . 2
10 19.41v 1779 . . . 4
11 simpr 103 . . . . 5
12 gencbvex.4 . . . . . . . 8
13 eqcom 2039 . . . . . . . . . . 11
1413biimpi 113 . . . . . . . . . 10
1514adantl 262 . . . . . . . . 9
1615eximi 1488 . . . . . . . 8
1712, 16sylbi 114 . . . . . . 7
1817adantr 261 . . . . . 6
1918ancri 307 . . . . 5
2011, 19impbii 117 . . . 4
2110, 20bitri 173 . . 3
2221exbii 1493 . 2
231, 9, 223bitr3i 199 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   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-5 1333  ax-7 1334  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:  gencbvex2  2595
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