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Theorem exdistrf 2035
Description: Distribution of existential quantifiers, with a bound-variable hypothesis saying that  y is not free in  ph, but  x can be free in  ph (and there is no distinct variable condition on  x and  y). (Contributed by Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.)
Hypothesis
Ref Expression
exdistrf.1  |-  ( -. 
A. x  x  =  y  ->  F/ y ph )
Assertion
Ref Expression
exdistrf  |-  ( E. x E. y (
ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
)

Proof of Theorem exdistrf
StepHypRef Expression
1 nfe1 1780 . 2  |-  F/ x E. x ( ph  /\  E. y ps )
2 19.8a 1797 . . . . . 6  |-  ( ps 
->  E. y ps )
32anim2i 569 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ph  /\  E. y ps ) )
43eximi 1626 . . . 4  |-  ( E. y ( ph  /\  ps )  ->  E. y
( ph  /\  E. y ps ) )
5 biidd 237 . . . . 5  |-  ( A. x  x  =  y  ->  ( ( ph  /\  E. y ps )  <->  ( ph  /\ 
E. y ps )
) )
65drex1 2029 . . . 4  |-  ( A. x  x  =  y  ->  ( E. x (
ph  /\  E. y ps )  <->  E. y ( ph  /\ 
E. y ps )
) )
74, 6syl5ibr 221 . . 3  |-  ( A. x  x  =  y  ->  ( E. y (
ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
) )
8 19.40 1647 . . . 4  |-  ( E. y ( ph  /\  ps )  ->  ( E. y ph  /\  E. y ps ) )
9 exdistrf.1 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ y ph )
10919.9d 1831 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( E. y ph  ->  ph ) )
1110anim1d 564 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( ( E. y ph  /\  E. y ps )  ->  ( ph  /\  E. y ps ) ) )
12 19.8a 1797 . . . 4  |-  ( (
ph  /\  E. y ps )  ->  E. x
( ph  /\  E. y ps ) )
138, 11, 12syl56 34 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( E. y ( ph  /\  ps )  ->  E. x
( ph  /\  E. y ps ) ) )
147, 13pm2.61i 164 . 2  |-  ( E. y ( ph  /\  ps )  ->  E. x
( ph  /\  E. y ps ) )
151, 14exlimi 1850 1  |-  ( E. x E. y (
ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1368   E.wex 1587   F/wnf 1590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  oprabid  6227
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