Theorem xpeq12d 4629

Description: Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)

Hypotheses

Ref	Expression
xpeq1d.1	|- ( ph -> A = B )
xpeq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
xpeq12d	|- ( ph -> ( A X. C ) = ( B X. D ) )

Proof of Theorem xpeq12d

Step	Hyp	Ref	Expression
1		xpeq1d.1	. 2 |- ( ph -> A = B )
2		xpeq12d.2	. 2 |- ( ph -> C = D )
3		xpeq12 4623	. 2 |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )
4	1, 2, 3	syl2anc 409	1 |- ( ph -> ( A X. C ) = ( B X. D ) )

Syntax hints:   -> wi 4   = wceq 1343   X. cxp 4602

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-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-opab 4044  df-xp 4610

This theorem is referenced by:  sqxpeqd  4630  opeliunxp  4659  mpomptsx  6165  dmmpossx  6167  fmpox  6168  disjxp1  6204  erssxp  6524  cc2lem  7207  cc2  7208  fsum2dlemstep  11375  fisumcom2  11379  fprod2dlemstep  11563  fprodcom2fi  11567  txbas  12908