Theorem xpeq12d 4774

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 4768	. 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 1398   X. cxp 4747

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-opab 4172  df-xp 4755

This theorem is referenced by:  sqxpeqd  4775  opeliunxp  4805  mpomptsx  6393  dmmpossx  6395  fmpox  6396  disjxp1  6432  erssxp  6790  cc2lem  7580  cc2  7581  fsum2dlemstep  12120  fisumcom2  12124  fprod2dlemstep  12308  fprodcom2fi  12312  psrval  14814  txbas  15123