Theorem xpeq1 4798

Description: Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
xpeq1	|- (A = B -> (A X. C) = (B X. C))

Proof of Theorem xpeq1

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

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . 4 |- (A = B -> (x e. A <-> x e. B))
2	1	anbi1d 685	. . 3 |- (A = B -> ((x e. A /\ y e. C) <-> (x e. B /\ y e. C)))
3	2	opabbidv 4625	. 2 |- (A = B -> {<.x, y>. | (x e. A /\ y e. C)} = {<.x, y>. | (x e. B /\ y e. C)})
4		df-xp 4784	. 2 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
5		df-xp 4784	. 2 |- (B X. C) = {<.x, y>. | (x e. B /\ y e. C)}
6	3, 4, 5	3eqtr4g 2410	1 |- (A = B -> (A X. C) = (B X. C))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  {copab 4622   X. cxp 4770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-opab 4623  df-xp 4784

This theorem is referenced by:  xpeq12  4803  xpeq1i  4804  xpeq1d  4807  dmxpid  4924  reseq2  4929  xpnz  5045  xpdisj1  5047  xpcan2  5058  ovcross  5845  pmvalg  6010  xpsneng  6050  xpcomeng  6053