Theorem xpeq1d 4686

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 4677	. 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 1364   X. cxp 4661

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 4095  df-xp 4669

This theorem is referenced by:  xpssres  4981  ixpsnf1o  6795  xpfi  6993  hashxp  10918  psrval  14220  dvmptc  14953