Theorem xpeq1d 4627

Description: Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)

Hypothesis

Ref	Expression
xpeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
xpeq1d	|- ( ph -> ( A X. C ) = ( B X. C ) )

Proof of Theorem xpeq1d

Step	Hyp	Ref	Expression
1		xpeq1d.1	. 2 |- ( ph -> A = B )
2		xpeq1 4618	. 2 |- ( A = B -> ( A X. C ) = ( B X. C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A X. C ) = ( B X. C ) )

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:  xpssres  4919  ixpsnf1o  6702  xpfi  6895  hashxp  10739