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Theorem ixpeq2 7544
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 7543 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  X_ x  e.  A  B  C_  X_ x  e.  A  C )
2 ss2ixp 7543 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  X_ x  e.  A  C  C_  X_ x  e.  A  B )
31, 2anim12i 568 . 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 3485 . . . 4  |-  ( B  =  C  <->  ( B  C_  C  /\  C  C_  B ) )
54ralbii 2863 . . 3  |-  ( A. x  e.  A  B  =  C  <->  A. x  e.  A  ( B  C_  C  /\  C  C_  B ) )
6 r19.26 2962 . . 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 252 . 2  |-  ( A. x  e.  A  B  =  C  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B ) )
8 eqss 3485 . 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 269 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 370    = wceq 1437   A.wral 2782    C_ wss 3442   X_cixp 7530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-in 3449  df-ss 3456  df-ixp 7531
This theorem is referenced by:  ixpeq2dva  7545  ixpint  7557  prdsbas3  15338  pwsbas  15344  ptbasfi  20527  ptunimpt  20541  pttopon  20542  ptcld  20559  ptrescn  20585  ptuncnv  20753  ptunhmeo  20754  ptrest  31643  prdstotbnd  31830
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