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Theorem 2mos 2235
Description: Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
Hypothesis
Ref Expression
2mos.1  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
2mos  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
Distinct variable groups:    z, w, ph    x, y, ps    x, z, w, y
Allowed substitution hints:    ph( x, y)    ps( z, w)

Proof of Theorem 2mos
StepHypRef Expression
1 2mo 2234 . 2  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
[ z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
) )
2 nfv 1609 . . . . . . 7  |-  F/ x ps
3 nfv 1609 . . . . . . . . . 10  |-  F/ y  x  =  z
43sbrim 2020 . . . . . . . . 9  |-  ( [ w  /  y ] ( x  =  z  ->  ph )  <->  ( x  =  z  ->  [ w  /  y ] ph ) )
5 nfv 1609 . . . . . . . . . 10  |-  F/ y ( x  =  z  ->  ps )
6 2mos.1 . . . . . . . . . . . 12  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
76expcom 424 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
x  =  z  -> 
( ph  <->  ps ) ) )
87pm5.74d 238 . . . . . . . . . 10  |-  ( y  =  w  ->  (
( x  =  z  ->  ph )  <->  ( x  =  z  ->  ps )
) )
95, 8sbie 1991 . . . . . . . . 9  |-  ( [ w  /  y ] ( x  =  z  ->  ph )  <->  ( x  =  z  ->  ps )
)
104, 9bitr3i 242 . . . . . . . 8  |-  ( ( x  =  z  ->  [ w  /  y ] ph )  <->  ( x  =  z  ->  ps )
)
1110pm5.74ri 237 . . . . . . 7  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  ps ) )
122, 11sbie 1991 . . . . . 6  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  ps )
1312anbi2i 675 . . . . 5  |-  ( (
ph  /\  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( ph  /\  ps ) )
1413imbi1i 315 . . . 4  |-  ( ( ( ph  /\  [
z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
)  <->  ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
15142albii 1557 . . 3  |-  ( A. z A. w ( (
ph  /\  [ z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w ) )  <->  A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
16152albii 1557 . 2  |-  ( A. x A. y A. z A. w ( ( ph  /\ 
[ z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
)  <->  A. x A. y A. z A. w ( ( ph  /\  ps )  ->  ( x  =  z  /\  y  =  w ) ) )
171, 16bitri 240 1  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632   [wsb 1638
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  df-sb 1639
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