MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  exdistrf Unicode version

Theorem exdistrf 1924
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.)
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 biidd 228 . . . . 5  |-  ( A. x  x  =  y  ->  ( ( ph  /\  ps )  <->  ( ph  /\  ps ) ) )
21drex1 1920 . . . 4  |-  ( A. x  x  =  y  ->  ( E. x (
ph  /\  ps )  <->  E. y ( ph  /\  ps ) ) )
32drex2 1921 . . 3  |-  ( A. x  x  =  y  ->  ( E. x E. x ( ph  /\  ps )  <->  E. x E. y
( ph  /\  ps )
) )
4 nfe1 1718 . . . . 5  |-  F/ x E. x ( ph  /\  ps )
5419.9 1795 . . . 4  |-  ( E. x E. x (
ph  /\  ps )  <->  E. x ( ph  /\  ps ) )
6 19.8a 1730 . . . . . 6  |-  ( ps 
->  E. y ps )
76anim2i 552 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ph  /\  E. y ps ) )
87eximi 1566 . . . 4  |-  ( E. x ( ph  /\  ps )  ->  E. x
( ph  /\  E. y ps ) )
95, 8sylbi 187 . . 3  |-  ( E. x E. x (
ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
)
103, 9syl6bir 220 . 2  |-  ( A. x  x  =  y  ->  ( E. x E. y ( ph  /\  ps )  ->  E. x
( ph  /\  E. y ps ) ) )
11 nfnae 1909 . . 3  |-  F/ x  -.  A. x  x  =  y
12 19.40 1599 . . . 4  |-  ( E. y ( ph  /\  ps )  ->  ( E. y ph  /\  E. y ps ) )
13 exdistrf.1 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ y ph )
141319.9d 1796 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( E. y ph  ->  ph ) )
1514anim1d 547 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( ( E. y ph  /\  E. y ps )  ->  ( ph  /\  E. y ps ) ) )
1612, 15syl5 28 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( E. y ( ph  /\  ps )  ->  ( ph  /\ 
E. y ps )
) )
1711, 16eximd 1762 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. x E. y ( ph  /\ 
ps )  ->  E. x
( ph  /\  E. y ps ) ) )
1810, 17pm2.61i 156 1  |-  ( E. x E. y (
ph  /\  ps )  ->  E. x ( ph  /\ 
E. y ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531   F/wnf 1534
This theorem is referenced by:  oprabid  5898
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-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
  Copyright terms: Public domain W3C validator