Theorem xpeq1d 4500

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 4491	. 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 1299   X. cxp 4475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-opab 3930  df-xp 4483

This theorem is referenced by:  xpssres  4790  ixpsnf1o  6560  xpfi  6747  hashxp  10413