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Theorem gencbvex 2998
 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:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()   ()

Proof of Theorem gencbvex
StepHypRef Expression
1 excom 1756 . 2
2 gencbvex.1 . . . 4
3 gencbvex.3 . . . . . . 7
4 gencbvex.2 . . . . . . 7
53, 4anbi12d 692 . . . . . 6
65bicomd 193 . . . . 5
76eqcoms 2439 . . . 4
82, 7ceqsexv 2991 . . 3
98exbii 1592 . 2
10 19.41v 1924 . . . 4
11 simpr 448 . . . . 5
12 gencbvex.4 . . . . . . . 8
13 eqcom 2438 . . . . . . . . . . 11
1413biimpi 187 . . . . . . . . . 10
1514adantl 453 . . . . . . . . 9
1615eximi 1585 . . . . . . . 8
1712, 16sylbi 188 . . . . . . 7
1817adantr 452 . . . . . 6
1918ancri 536 . . . . 5
2011, 19impbii 181 . . . 4
2110, 20bitri 241 . . 3
2221exbii 1592 . 2
231, 9, 223bitr3i 267 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725  cvv 2956 This theorem is referenced by:  gencbvex2  2999  gencbval  3000 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-v 2958
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