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Theorem ixpeq2 7380
Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
Assertion
Ref Expression
ixpeq2  |-  ( A. x  e.  A  B  =  C  ->  X_ x  e.  A  B  =  X_ x  e.  A  C
)

Proof of Theorem ixpeq2
StepHypRef Expression
1 ss2ixp 7379 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  X_ x  e.  A  B  C_  X_ x  e.  A  C )
2 ss2ixp 7379 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  X_ x  e.  A  C  C_  X_ x  e.  A  B )
31, 2anim12i 566 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B )  ->  ( X_ x  e.  A  B  C_  X_ x  e.  A  C  /\  X_ x  e.  A  C  C_  X_ x  e.  A  B ) )
4 eqss 3472 . . . 4  |-  ( B  =  C  <->  ( B  C_  C  /\  C  C_  B ) )
54ralbii 2834 . . 3  |-  ( A. x  e.  A  B  =  C  <->  A. x  e.  A  ( B  C_  C  /\  C  C_  B ) )
6 r19.26 2948 . . 3  |-  ( A. x  e.  A  ( B  C_  C  /\  C  C_  B )  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B
) )
75, 6bitri 249 . 2  |-  ( A. x  e.  A  B  =  C  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B ) )
8 eqss 3472 . 2  |-  ( X_ x  e.  A  B  =  X_ x  e.  A  C 
<->  ( X_ x  e.  A  B  C_  X_ x  e.  A  C  /\  X_ x  e.  A  C  C_  X_ x  e.  A  B ) )
93, 7, 83imtr4i 266 1  |-  ( A. x  e.  A  B  =  C  ->  X_ x  e.  A  B  =  X_ x  e.  A  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   A.wral 2795    C_ wss 3429   X_cixp 7366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-in 3436  df-ss 3443  df-ixp 7367
This theorem is referenced by:  ixpeq2dva  7381  ixpint  7393  prdsbas3  14530  pwsbas  14536  ptbasfi  19279  ptunimpt  19293  pttopon  19294  ptcld  19311  ptrescn  19337  ptuncnv  19505  ptunhmeo  19506  ptrest  28566  prdstotbnd  28834
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