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Theorem ixpeq2dva 7474
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dva.1  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
ixpeq2dva  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem ixpeq2dva
StepHypRef Expression
1 ixpeq2dva.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
21ralrimiva 2871 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 ixpeq2 7473 . 2  |-  ( A. x  e.  A  B  =  C  ->  X_ x  e.  A  B  =  X_ x  e.  A  C
)
42, 3syl 16 1  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   X_cixp 7459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-in 3476  df-ss 3483  df-ixp 7460
This theorem is referenced by:  ixpeq2dv  7475  dfac9  8505  xpsfrn2  14814  xpslem  14817  funcpropd  15116  natpropd  15192  prdsmgp  17036  frlmip  18569  elptr2  19803  dfac14  19847  xkoptsub  19883  prdsxmslem2  20760  rrxip  21550  ptrest  29476  prdsbnd2  29745
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