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Theorem gencbvex 3153
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
gencbvex.2
gencbvex.3
gencbvex.4
Assertion
Ref Expression
gencbvex
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem gencbvex
StepHypRef Expression
1 excom 1849 . 2
2 gencbvex.1 . . . 4
3 gencbvex.3 . . . . . . 7
4 gencbvex.2 . . . . . . 7
53, 4anbi12d 710 . . . . . 6
65bicomd 201 . . . . 5
76eqcoms 2469 . . . 4
82, 7ceqsexv 3146 . . 3
98exbii 1667 . 2
10 19.41v 1771 . . . 4
11 simpr 461 . . . . 5
12 gencbvex.4 . . . . . . . 8
13 eqcom 2466 . . . . . . . . . . 11
1413biimpi 194 . . . . . . . . . 10
1514adantl 466 . . . . . . . . 9
1615eximi 1656 . . . . . . . 8
1712, 16sylbi 195 . . . . . . 7
1817adantr 465 . . . . . 6
1918ancri 552 . . . . 5
2011, 19impbii 188 . . . 4
2110, 20bitri 249 . . 3
2221exbii 1667 . 2
231, 9, 223bitr3i 275 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109
This theorem is referenced by:  gencbvex2  3154  gencbval  3155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111
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