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Theorem 2sb5rf 2290
Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove distinct variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.)
Hypotheses
Ref Expression
2sb5rf.1  |-  F/ z
ph
2sb5rf.2  |-  F/ w ph
Assertion
Ref Expression
2sb5rf  |-  ( ph  <->  E. z E. w ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [
w  /  y ]
ph ) )
Distinct variable group:    z, w
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem 2sb5rf
StepHypRef Expression
1 2sb5rf.2 . . . . 5  |-  F/ w ph
2119.41 2061 . . . 4  |-  ( E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  ( E. w
( z  =  x  /\  w  =  y )  /\  ph )
)
32exbii 1728 . . 3  |-  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  E. z ( E. w
( z  =  x  /\  w  =  y )  /\  ph )
)
4 2sb5rf.1 . . . 4  |-  F/ z
ph
5419.41 2061 . . 3  |-  ( E. z ( E. w
( z  =  x  /\  w  =  y )  /\  ph )  <->  ( E. z E. w
( z  =  x  /\  w  =  y )  /\  ph )
)
63, 5bitri 257 . 2  |-  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  ( E. z E. w
( z  =  x  /\  w  =  y )  /\  ph )
)
7 sbequ12r 2094 . . . . 5  |-  ( z  =  x  ->  ( [ z  /  x ] [ w  /  y ] ph  <->  [ w  /  y ] ph ) )
8 sbequ12r 2094 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  y ] ph  <->  ph ) )
97, 8sylan9bb 711 . . . 4  |-  ( ( z  =  x  /\  w  =  y )  ->  ( [ z  /  x ] [ w  / 
y ] ph  <->  ph ) )
109pm5.32i 647 . . 3  |-  ( ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [
w  /  y ]
ph )  <->  ( (
z  =  x  /\  w  =  y )  /\  ph ) )
11102exbii 1729 . 2  |-  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [
w  /  y ]
ph )  <->  E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph ) )
12 2ax6e 2289 . . 3  |-  E. z E. w ( z  =  x  /\  w  =  y )
1312biantrur 513 . 2  |-  ( ph  <->  ( E. z E. w
( z  =  x  /\  w  =  y )  /\  ph )
)
146, 11, 133bitr4ri 286 1  |-  ( ph  <->  E. z E. w ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [
w  /  y ]
ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375   E.wex 1673   F/wnf 1677   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  sbel2x  2298
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