MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mo3OLD Structured version   Unicode version

Theorem mo3OLD 2325
Description: Obsolete proof of mo3 2324 as of 20-Jul-2019. (Contributed by NM, 8-Mar-1995.) (Revised by Wolf Lammen, 3-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
mo3.1  |-  F/ y
ph
Assertion
Ref Expression
mo3OLD  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem mo3OLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mo2v 2291 . . 3  |-  ( E* x ph  <->  E. z A. x ( ph  ->  x  =  z ) )
2 mo3.1 . . . . . . . . 9  |-  F/ y
ph
3 nfv 1712 . . . . . . . . 9  |-  F/ y  x  =  z
42, 3nfim 1925 . . . . . . . 8  |-  F/ y ( ph  ->  x  =  z )
5 nfs1v 2183 . . . . . . . . 9  |-  F/ x [ y  /  x ] ph
6 nfv 1712 . . . . . . . . 9  |-  F/ x  y  =  z
75, 6nfim 1925 . . . . . . . 8  |-  F/ x
( [ y  /  x ] ph  ->  y  =  z )
8 sbequ2 1746 . . . . . . . . 9  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
9 ax-7 1795 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
108, 9imim12d 74 . . . . . . . 8  |-  ( x  =  y  ->  (
( ph  ->  x  =  z )  ->  ( [ y  /  x ] ph  ->  y  =  z ) ) )
114, 7, 10cbv3 2020 . . . . . . 7  |-  ( A. x ( ph  ->  x  =  z )  ->  A. y ( [ y  /  x ] ph  ->  y  =  z ) )
1211ancli 549 . . . . . 6  |-  ( A. x ( ph  ->  x  =  z )  -> 
( A. x (
ph  ->  x  =  z )  /\  A. y
( [ y  /  x ] ph  ->  y  =  z ) ) )
134, 7aaan 1980 . . . . . 6  |-  ( A. x A. y ( (
ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) )  <->  ( A. x ( ph  ->  x  =  z )  /\  A. y ( [ y  /  x ] ph  ->  y  =  z ) ) )
1412, 13sylibr 212 . . . . 5  |-  ( A. x ( ph  ->  x  =  z )  ->  A. x A. y ( ( ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) ) )
15 prth 569 . . . . . . 7  |-  ( ( ( ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  ( x  =  z  /\  y  =  z ) ) )
16 equtr2 1807 . . . . . . 7  |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
1715, 16syl6 33 . . . . . 6  |-  ( ( ( ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
18172alimi 1639 . . . . 5  |-  ( A. x A. y ( (
ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) )  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
1914, 18syl 16 . . . 4  |-  ( A. x ( ph  ->  x  =  z )  ->  A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )
2019exlimiv 1727 . . 3  |-  ( E. z A. x (
ph  ->  x  =  z )  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
211, 20sylbi 195 . 2  |-  ( E* x ph  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
22 nfa1 1902 . . . . . 6  |-  F/ y A. y A. x
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y )
23 pm3.3 442 . . . . . . . . . 10  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  ( [ y  /  x ] ph  ->  x  =  y ) ) )
2423com3r 79 . . . . . . . . 9  |-  ( [ y  /  x ] ph  ->  ( ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  x  =  y ) ) )
255, 24alimd 1881 . . . . . . . 8  |-  ( [ y  /  x ] ph  ->  ( A. x
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  A. x
( ph  ->  x  =  y ) ) )
2625com12 31 . . . . . . 7  |-  ( A. x ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( [ y  /  x ] ph  ->  A. x
( ph  ->  x  =  y ) ) )
2726sps 1870 . . . . . 6  |-  ( A. y A. x ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( [ y  /  x ] ph  ->  A. x ( ph  ->  x  =  y ) ) )
2822, 27eximd 1887 . . . . 5  |-  ( A. y A. x ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( E. y [ y  /  x ] ph  ->  E. y A. x ( ph  ->  x  =  y ) ) )
292sb8e 2170 . . . . 5  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
302mo2 2295 . . . . 5  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
3128, 29, 303imtr4g 270 . . . 4  |-  ( A. y A. x ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( E. x ph  ->  E* x ph ) )
32 moabs 2317 . . . 4  |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
3331, 32sylibr 212 . . 3  |-  ( A. y A. x ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  E* x ph )
3433alcoms 1848 . 2  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  E* x ph )
3521, 34impbii 188 1  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396   E.wex 1617   F/wnf 1621   [wsb 1744   E*wmo 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator