Theorem xpeq12d 4689

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 4683	. 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 1364   X. cxp 4662

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-opab 4096  df-xp 4670

This theorem is referenced by:  sqxpeqd  4690  opeliunxp  4719  mpomptsx  6264  dmmpossx  6266  fmpox  6267  disjxp1  6303  erssxp  6624  cc2lem  7349  cc2  7350  fsum2dlemstep  11616  fisumcom2  11620  fprod2dlemstep  11804  fprodcom2fi  11808  psrval  14296  txbas  14578