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Theorem reu2eqd 3235
Description: Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.)
Hypotheses
Ref Expression
reu2eqd.1  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
reu2eqd.2  |-  ( x  =  C  ->  ( ps 
<->  th ) )
reu2eqd.3  |-  ( ph  ->  E! x  e.  A  ps )
reu2eqd.4  |-  ( ph  ->  B  e.  A )
reu2eqd.5  |-  ( ph  ->  C  e.  A )
reu2eqd.6  |-  ( ph  ->  ch )
reu2eqd.7  |-  ( ph  ->  th )
Assertion
Ref Expression
reu2eqd  |-  ( ph  ->  B  =  C )
Distinct variable groups:    x, A    x, B    x, C    ch, x    th, x
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem reu2eqd
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 reu2eqd.6 . 2  |-  ( ph  ->  ch )
2 reu2eqd.7 . 2  |-  ( ph  ->  th )
3 reu2eqd.3 . . . . 5  |-  ( ph  ->  E! x  e.  A  ps )
4 reu2 3226 . . . . 5  |-  ( E! x  e.  A  ps  <->  ( E. x  e.  A  ps  /\  A. x  e.  A  A. y  e.  A  ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
) )
53, 4sylib 200 . . . 4  |-  ( ph  ->  ( E. x  e.  A  ps  /\  A. x  e.  A  A. y  e.  A  (
( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
65simprd 465 . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y ) )
7 reu2eqd.4 . . . 4  |-  ( ph  ->  B  e.  A )
8 reu2eqd.5 . . . 4  |-  ( ph  ->  C  e.  A )
9 nfv 1761 . . . . . . 7  |-  F/ x ch
10 nfs1v 2266 . . . . . . 7  |-  F/ x [ y  /  x ] ps
119, 10nfan 2011 . . . . . 6  |-  F/ x
( ch  /\  [
y  /  x ] ps )
12 nfv 1761 . . . . . 6  |-  F/ x  B  =  y
1311, 12nfim 2003 . . . . 5  |-  F/ x
( ( ch  /\  [ y  /  x ] ps )  ->  B  =  y )
14 nfv 1761 . . . . 5  |-  F/ y ( ( ch  /\  th )  ->  B  =  C )
15 reu2eqd.1 . . . . . . 7  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
1615anbi1d 711 . . . . . 6  |-  ( x  =  B  ->  (
( ps  /\  [
y  /  x ] ps )  <->  ( ch  /\  [ y  /  x ] ps ) ) )
17 eqeq1 2455 . . . . . 6  |-  ( x  =  B  ->  (
x  =  y  <->  B  =  y ) )
1816, 17imbi12d 322 . . . . 5  |-  ( x  =  B  ->  (
( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y )  <->  ( ( ch  /\  [ y  /  x ] ps )  ->  B  =  y )
) )
19 nfv 1761 . . . . . . . 8  |-  F/ x th
20 reu2eqd.2 . . . . . . . 8  |-  ( x  =  C  ->  ( ps 
<->  th ) )
2119, 20sbhypf 3095 . . . . . . 7  |-  ( y  =  C  ->  ( [ y  /  x ] ps  <->  th ) )
2221anbi2d 710 . . . . . 6  |-  ( y  =  C  ->  (
( ch  /\  [
y  /  x ] ps )  <->  ( ch  /\  th ) ) )
23 eqeq2 2462 . . . . . 6  |-  ( y  =  C  ->  ( B  =  y  <->  B  =  C ) )
2422, 23imbi12d 322 . . . . 5  |-  ( y  =  C  ->  (
( ( ch  /\  [ y  /  x ] ps )  ->  B  =  y )  <->  ( ( ch  /\  th )  ->  B  =  C )
) )
2513, 14, 18, 24rspc2 3158 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( A. x  e.  A  A. y  e.  A  ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )  ->  ( ( ch  /\  th )  ->  B  =  C ) ) )
267, 8, 25syl2anc 667 . . 3  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  A  ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )  ->  ( ( ch  /\  th )  ->  B  =  C ) ) )
276, 26mpd 15 . 2  |-  ( ph  ->  ( ( ch  /\  th )  ->  B  =  C ) )
281, 2, 27mp2and 685 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444   [wsb 1797    e. wcel 1887   A.wral 2737   E.wrex 2738   E!wreu 2739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-reu 2744  df-v 3047
This theorem is referenced by:  qtophmeo  20832  footeq  24766  mideulem2  24776  lmieq  24833
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