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Theorem gencbvex 2843
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  |-  A  e. 
_V
gencbvex.2  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
gencbvex.3  |-  ( A  =  y  ->  ( ch 
<->  th ) )
gencbvex.4  |-  ( th  <->  E. x ( ch  /\  A  =  y )
)
Assertion
Ref Expression
gencbvex  |-  ( E. x ( ch  /\  ph )  <->  E. y ( th 
/\  ps ) )
Distinct variable groups:    ps, x    ph, y    th, x    ch, y    y, A
Allowed substitution hints:    ph( x)    ps( y)    ch( x)    th( y)    A( x)

Proof of Theorem gencbvex
StepHypRef Expression
1 excom 1798 . 2  |-  ( E. x E. y ( y  =  A  /\  ( th  /\  ps )
)  <->  E. y E. x
( y  =  A  /\  ( th  /\  ps ) ) )
2 gencbvex.1 . . . 4  |-  A  e. 
_V
3 gencbvex.3 . . . . . . 7  |-  ( A  =  y  ->  ( ch 
<->  th ) )
4 gencbvex.2 . . . . . . 7  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
53, 4anbi12d 691 . . . . . 6  |-  ( A  =  y  ->  (
( ch  /\  ph ) 
<->  ( th  /\  ps ) ) )
65bicomd 192 . . . . 5  |-  ( A  =  y  ->  (
( th  /\  ps ) 
<->  ( ch  /\  ph ) ) )
76eqcoms 2299 . . . 4  |-  ( y  =  A  ->  (
( th  /\  ps ) 
<->  ( ch  /\  ph ) ) )
82, 7ceqsexv 2836 . . 3  |-  ( E. y ( y  =  A  /\  ( th 
/\  ps ) )  <->  ( ch  /\ 
ph ) )
98exbii 1572 . 2  |-  ( E. x E. y ( y  =  A  /\  ( th  /\  ps )
)  <->  E. x ( ch 
/\  ph ) )
10 19.41v 1854 . . . 4  |-  ( E. x ( y  =  A  /\  ( th 
/\  ps ) )  <->  ( E. x  y  =  A  /\  ( th  /\  ps ) ) )
11 simpr 447 . . . . 5  |-  ( ( E. x  y  =  A  /\  ( th 
/\  ps ) )  -> 
( th  /\  ps ) )
12 gencbvex.4 . . . . . . . 8  |-  ( th  <->  E. x ( ch  /\  A  =  y )
)
13 eqcom 2298 . . . . . . . . . . 11  |-  ( A  =  y  <->  y  =  A )
1413biimpi 186 . . . . . . . . . 10  |-  ( A  =  y  ->  y  =  A )
1514adantl 452 . . . . . . . . 9  |-  ( ( ch  /\  A  =  y )  ->  y  =  A )
1615eximi 1566 . . . . . . . 8  |-  ( E. x ( ch  /\  A  =  y )  ->  E. x  y  =  A )
1712, 16sylbi 187 . . . . . . 7  |-  ( th 
->  E. x  y  =  A )
1817adantr 451 . . . . . 6  |-  ( ( th  /\  ps )  ->  E. x  y  =  A )
1918ancri 535 . . . . 5  |-  ( ( th  /\  ps )  ->  ( E. x  y  =  A  /\  ( th  /\  ps ) ) )
2011, 19impbii 180 . . . 4  |-  ( ( E. x  y  =  A  /\  ( th 
/\  ps ) )  <->  ( th  /\  ps ) )
2110, 20bitri 240 . . 3  |-  ( E. x ( y  =  A  /\  ( th 
/\  ps ) )  <->  ( th  /\  ps ) )
2221exbii 1572 . 2  |-  ( E. y E. x ( y  =  A  /\  ( th  /\  ps )
)  <->  E. y ( th 
/\  ps ) )
231, 9, 223bitr3i 266 1  |-  ( E. x ( ch  /\  ph )  <->  E. y ( th 
/\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  gencbvex2  2844  gencbval  2845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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