Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1316 Structured version   Unicode version

Theorem bnj1316 32967
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 2619 . . . 4  |-  F/_ x A
3 bnj1316.2 . . . . 5  |-  ( y  e.  B  ->  A. x  y  e.  B )
43nfcii 2619 . . . 4  |-  F/_ x B
52, 4nfeq 2640 . . 3  |-  F/ x  A  =  B
65nfri 1822 . 2  |-  ( A  =  B  ->  A. x  A  =  B )
76bnj956 32923 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 1377    = wceq 1379    e. wcel 1767   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-iun 4327
This theorem is referenced by:  bnj1000  33087  bnj1318  33169
  Copyright terms: Public domain W3C validator