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Theorem bnj1316 29266
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 2565 . . . 4  |-  F/_ x A
3 bnj1316.2 . . . . 5  |-  ( y  e.  B  ->  A. x  y  e.  B )
43nfcii 2565 . . . 4  |-  F/_ x B
52, 4nfeq 2581 . . 3  |-  F/ x  A  =  B
65nfri 1779 . 2  |-  ( A  =  B  ->  A. x  A  =  B )
76bnj956 29221 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 1550    = wceq 1653    e. wcel 1726   U_ciun 4095
This theorem is referenced by:  bnj1000  29386  bnj1318  29468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-iun 4097
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