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Theorem 2sb6rf 2251
Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove 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
2sb6rf  |-  ( ph  <->  A. z A. 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 2sb6rf
StepHypRef Expression
1 sbequ12r 2052 . . . . 5  |-  ( z  =  x  ->  ( [ z  /  x ] [ w  /  y ] ph  <->  [ w  /  y ] ph ) )
2 sbequ12r 2052 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  y ] ph  <->  ph ) )
31, 2sylan9bb 704 . . . 4  |-  ( ( z  =  x  /\  w  =  y )  ->  ( [ z  /  x ] [ w  / 
y ] ph  <->  ph ) )
43pm5.74i 248 . . 3  |-  ( ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( ( z  =  x  /\  w  =  y )  ->  ph )
)
542albii 1686 . 2  |-  ( A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph )  <->  A. z A. w ( ( z  =  x  /\  w  =  y )  ->  ph ) )
6 2sb5rf.2 . . . . 5  |-  F/ w ph
7619.23 1970 . . . 4  |-  ( A. w ( ( z  =  x  /\  w  =  y )  ->  ph )  <->  ( E. w
( z  =  x  /\  w  =  y )  ->  ph ) )
87albii 1685 . . 3  |-  ( A. z A. w ( ( z  =  x  /\  w  =  y )  ->  ph )  <->  A. z
( E. w ( z  =  x  /\  w  =  y )  ->  ph ) )
9 2sb5rf.1 . . . 4  |-  F/ z
ph
10919.23 1970 . . 3  |-  ( A. z ( E. w
( z  =  x  /\  w  =  y )  ->  ph )  <->  ( E. z E. w ( z  =  x  /\  w  =  y )  ->  ph ) )
118, 10bitri 252 . 2  |-  ( A. z A. w ( ( z  =  x  /\  w  =  y )  ->  ph )  <->  ( E. z E. w ( z  =  x  /\  w  =  y )  ->  ph ) )
12 2ax6e 2249 . . 3  |-  E. z E. w ( z  =  x  /\  w  =  y )
13 pm5.5 337 . . 3  |-  ( E. z E. w ( z  =  x  /\  w  =  y )  ->  ( ( E. z E. w ( z  =  x  /\  w  =  y )  ->  ph )  <->  ph ) )
1412, 13ax-mp 5 . 2  |-  ( ( E. z E. w
( z  =  x  /\  w  =  y )  ->  ph )  <->  ph )
155, 11, 143bitrri 275 1  |-  ( ph  <->  A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1657   F/wnf 1661   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791
This theorem is referenced by: (None)
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