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Theorem cleqh 2572
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2637. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2104. (Revised by BJ, 30-Nov-2020.)
Hypotheses
Ref Expression
cleqh.1  |-  ( y  e.  A  ->  A. x  y  e.  A )
cleqh.2  |-  ( y  e.  B  ->  A. x  y  e.  B )
Assertion
Ref Expression
cleqh  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Distinct variable groups:    y, A    y, B    x, y
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem cleqh
StepHypRef Expression
1 dfcleq 2465 . 2  |-  ( A  =  B  <->  A. y
( y  e.  A  <->  y  e.  B ) )
2 nfv 1769 . . 3  |-  F/ y ( x  e.  A  <->  x  e.  B )
3 cleqh.1 . . . . 5  |-  ( y  e.  A  ->  A. x  y  e.  A )
43nfi 1682 . . . 4  |-  F/ x  y  e.  A
5 cleqh.2 . . . . 5  |-  ( y  e.  B  ->  A. x  y  e.  B )
65nfi 1682 . . . 4  |-  F/ x  y  e.  B
74, 6nfbi 2037 . . 3  |-  F/ x
( y  e.  A  <->  y  e.  B )
8 eleq1 2537 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
9 eleq1 2537 . . . 4  |-  ( x  =  y  ->  (
x  e.  B  <->  y  e.  B ) )
108, 9bibi12d 328 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  <->  x  e.  B )  <->  ( y  e.  A  <->  y  e.  B
) ) )
112, 7, 10cbvalv1 2083 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  B
)  <->  A. y ( y  e.  A  <->  y  e.  B ) )
121, 11bitr4i 260 1  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452    e. wcel 1904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-cleq 2464  df-clel 2467
This theorem is referenced by:  abeq2  2580  abbi  2585  cleqf  2637  abeq2f  2639  bj-abeq2  31454
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