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Theorem ixpeq2 7078
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 7077 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  X_ x  e.  A  B  C_  X_ x  e.  A  C )
2 ss2ixp 7077 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  X_ x  e.  A  C  C_  X_ x  e.  A  B )
31, 2anim12i 551 . 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 3365 . . . 4  |-  ( B  =  C  <->  ( B  C_  C  /\  C  C_  B ) )
54ralbii 2731 . . 3  |-  ( A. x  e.  A  B  =  C  <->  A. x  e.  A  ( B  C_  C  /\  C  C_  B ) )
6 r19.26 2840 . . 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 242 . 2  |-  ( A. x  e.  A  B  =  C  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B ) )
8 eqss 3365 . 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 259 1  |-  ( A. x  e.  A  B  =  C  ->  X_ x  e.  A  B  =  X_ x  e.  A  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653   A.wral 2707    C_ wss 3322   X_cixp 7065
This theorem is referenced by:  ixpeq2dva  7079  ixpint  7091  prdsbas3  13705  pwsbas  13711  ptbasfi  17615  ptunimpt  17629  pttopon  17630  ptcld  17647  ptrescn  17673  ptuncnv  17841  ptunhmeo  17842  prdstotbnd  26505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-in 3329  df-ss 3336  df-ixp 7066
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