Theorem xpeq2 4428

Description: Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)

Assertion

Ref	Expression
xpeq2	|- ( A = B -> ( C X. A ) = ( C X. B ) )

Proof of Theorem xpeq2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2148	. . . 4 |- ( A = B -> ( y e. A <-> y e. B ) )
2	1	anbi2d 452	. . 3 |- ( A = B -> ( ( x e. C /\ y e. A ) <-> ( x e. C /\ y e. B ) ) )
3	2	opabbidv 3881	. 2 |- ( A = B -> { <. x , y >. | ( x e. C /\ y e. A ) } = { <. x , y >. | ( x e. C /\ y e. B ) } )
4		df-xp 4419	. 2 |- ( C X. A ) = { <. x , y >. | ( x e. C /\ y e. A ) }
5		df-xp 4419	. 2 |- ( C X. B ) = { <. x , y >. | ( x e. C /\ y e. B ) }
6	3, 4, 5	3eqtr4g 2142	1 |- ( A = B -> ( C X. A ) = ( C X. B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1287   e. wcel 1436   {copab 3875   X. cxp 4411

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-opab 3877  df-xp 4419

This theorem is referenced by:  xpeq12  4432  xpeq2i  4434  xpeq2d  4437  xpeq0r  4822  xpdisj2  4824  pmvalg  6370  xpcomeng  6498  djueq12  6679