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Theorem bnj1316 28910
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 2539 . . . 4  |-  F/_ x A
3 bnj1316.2 . . . . 5  |-  ( y  e.  B  ->  A. x  y  e.  B )
43nfcii 2539 . . . 4  |-  F/_ x B
52, 4nfeq 2555 . . 3  |-  F/ x  A  =  B
65nfri 1774 . 2  |-  ( A  =  B  ->  A. x  A  =  B )
76bnj956 28865 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 1546    = wceq 1649    e. wcel 1721   U_ciun 4061
This theorem is referenced by:  bnj1000  29030  bnj1318  29112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-iun 4063
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