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Theorem bnj1316 31814
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 2570 . . . 4  |-  F/_ x A
3 bnj1316.2 . . . . 5  |-  ( y  e.  B  ->  A. x  y  e.  B )
43nfcii 2570 . . . 4  |-  F/_ x B
52, 4nfeq 2586 . . 3  |-  F/ x  A  =  B
65nfri 1808 . 2  |-  ( A  =  B  ->  A. x  A  =  B )
76bnj956 31770 1  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1367    = wceq 1369    e. wcel 1756   U_ciun 4171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-iun 4173
This theorem is referenced by:  bnj1000  31934  bnj1318  32016
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