ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ixpeq1d Unicode version

Theorem ixpeq1d 6608
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ixpeq1d  |-  ( ph  -> 
X_ x  e.  A  C  =  X_ x  e.  B  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)

Proof of Theorem ixpeq1d
StepHypRef Expression
1 ixpeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ixpeq1 6607 . 2  |-  ( A  =  B  ->  X_ x  e.  A  C  =  X_ x  e.  B  C
)
31, 2syl 14 1  |-  ( ph  -> 
X_ x  e.  A  C  =  X_ x  e.  B  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   X_cixp 6596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-fn 5130  df-ixp 6597
This theorem is referenced by:  elixpsn  6633  ixpsnf1o  6634
  Copyright terms: Public domain W3C validator