Theorem xpeq12d 4744

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 4738	. 2 |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A X. C ) = ( B X. D ) )

Syntax hints:   -> wi 4   = wceq 1395   X. cxp 4717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-opab 4146  df-xp 4725

This theorem is referenced by:  sqxpeqd  4745  opeliunxp  4774  mpomptsx  6349  dmmpossx  6351  fmpox  6352  disjxp1  6388  erssxp  6711  cc2lem  7463  cc2  7464  fsum2dlemstep  11960  fisumcom2  11964  fprod2dlemstep  12148  fprodcom2fi  12152  psrval  14645  txbas  14947