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Theorem exdistrfor 1728
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 Jim Kingdon, 25-Feb-2018.)
Hypothesis
Ref Expression
exdistrfor.1  |-  ( A. x  x  =  y  \/  A. x F/ y
ph )
Assertion
Ref Expression
exdistrfor  |-  ( E. x E. y (
ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
)

Proof of Theorem exdistrfor
StepHypRef Expression
1 exdistrfor.1 . 2  |-  ( A. x  x  =  y  \/  A. x F/ y
ph )
2 biidd 170 . . . . . 6  |-  ( A. x  x  =  y  ->  ( ( ph  /\  ps )  <->  ( ph  /\  ps ) ) )
32drex1 1726 . . . . 5  |-  ( A. x  x  =  y  ->  ( E. x (
ph  /\  ps )  <->  E. y ( ph  /\  ps ) ) )
43drex2 1667 . . . 4  |-  ( A. x  x  =  y  ->  ( E. x E. x ( ph  /\  ps )  <->  E. x E. y
( ph  /\  ps )
) )
5 hbe1 1429 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  A. x E. x ( ph  /\  ps ) )
6519.9h 1579 . . . . 5  |-  ( E. x E. x (
ph  /\  ps )  <->  E. x ( ph  /\  ps ) )
7 19.8a 1527 . . . . . . 7  |-  ( ps 
->  E. y ps )
87anim2i 334 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ( ph  /\  E. y ps ) )
98eximi 1536 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  E. x
( ph  /\  E. y ps ) )
106, 9sylbi 119 . . . 4  |-  ( E. x E. x (
ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
)
114, 10syl6bir 162 . . 3  |-  ( A. x  x  =  y  ->  ( E. x E. y ( ph  /\  ps )  ->  E. x
( ph  /\  E. y ps ) ) )
12 ax-ial 1472 . . . 4  |-  ( A. x F/ y ph  ->  A. x A. x F/ y ph )
13 19.40 1567 . . . . . 6  |-  ( E. y ( ph  /\  ps )  ->  ( E. y ph  /\  E. y ps ) )
14 19.9t 1578 . . . . . . . 8  |-  ( F/ y ph  ->  ( E. y ph  <->  ph ) )
1514biimpd 142 . . . . . . 7  |-  ( F/ y ph  ->  ( E. y ph  ->  ph )
)
1615anim1d 329 . . . . . 6  |-  ( F/ y ph  ->  (
( E. y ph  /\ 
E. y ps )  ->  ( ph  /\  E. y ps ) ) )
1713, 16syl5 32 . . . . 5  |-  ( F/ y ph  ->  ( E. y ( ph  /\  ps )  ->  ( ph  /\ 
E. y ps )
) )
1817sps 1475 . . . 4  |-  ( A. x F/ y ph  ->  ( E. y ( ph  /\ 
ps )  ->  ( ph  /\  E. y ps ) ) )
1912, 18eximdh 1547 . . 3  |-  ( A. x F/ y ph  ->  ( E. x E. y
( ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
) )
2011, 19jaoi 671 . 2  |-  ( ( A. x  x  =  y  \/  A. x F/ y ph )  -> 
( E. x E. y ( ph  /\  ps )  ->  E. x
( ph  /\  E. y ps ) ) )
211, 20ax-mp 7 1  |-  ( E. x E. y (
ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664   A.wal 1287   F/wnf 1394   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  oprabidlem  5662
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