Theorem xpeq12d 4801

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 4795	. 2 |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> ( A X. C ) = ( B X. D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   X. cxp 4774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-opab 4194  df-xp 4782

This theorem is used by:  sqxpeqd  4802  opeliunxp  4832  mpomptsx  6437  dmmpossx  6439  fmpox  6440  disjxp1  6476  erssxp  6834  cc2lem  7638  cc2  7639  fsum2dlemstep  12226  fisumcom2  12230  fprod2dlemstep  12414  fprodcom2fi  12418  gsumcom2fi  14275  psrval  15159  txbas  15484