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Theorem bnj1316 28605
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1316.1  |-  ( y  e.  A  ->  A. x  y  e.  A )
bnj1316.2  |-  ( y  e.  B  ->  A. x  y  e.  B )
Assertion
Ref Expression
bnj1316  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
Distinct variable groups:    y, A    y, B    x, y
Allowed substitution hints:    A( x)    B( x)    C( x, y)

Proof of Theorem bnj1316
StepHypRef Expression
1 bnj1316.1 . . . . 5  |-  ( y  e.  A  ->  A. x  y  e.  A )
21nfcii 2493 . . . 4  |-  F/_ x A
3 bnj1316.2 . . . . 5  |-  ( y  e.  B  ->  A. x  y  e.  B )
43nfcii 2493 . . . 4  |-  F/_ x B
52, 4nfeq 2509 . . 3  |-  F/ x  A  =  B
65nfri 1768 . 2  |-  ( A  =  B  ->  A. x  A  =  B )
76bnj956 28560 1  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1545    = wceq 1647    e. wcel 1715   U_ciun 4007
This theorem is referenced by:  bnj1000  28725  bnj1318  28807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rex 2634  df-iun 4009
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