Theorem xpeq2 4800

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

Assertion

Ref	Expression
xpeq2	⊢ (A = B → (C × A) = (C × B))

Proof of Theorem xpeq2

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

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . 4 ⊢ (A = B → (y ∈ A ↔ y ∈ B))
2	1	anbi2d 684	. . 3 ⊢ (A = B → ((x ∈ C ∧ y ∈ A) ↔ (x ∈ C ∧ y ∈ B)))
3	2	opabbidv 4626	. 2 ⊢ (A = B → {⟨x, y⟩ ∣ (x ∈ C ∧ y ∈ A)} = {⟨x, y⟩ ∣ (x ∈ C ∧ y ∈ B)})
4		df-xp 4785	. 2 ⊢ (C × A) = {⟨x, y⟩ ∣ (x ∈ C ∧ y ∈ A)}
5		df-xp 4785	. 2 ⊢ (C × B) = {⟨x, y⟩ ∣ (x ∈ C ∧ y ∈ B)}
6	3, 4, 5	3eqtr4g 2410	1 ⊢ (A = B → (C × A) = (C × B))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {copab 4623   × cxp 4771

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 4624  df-xp 4785

This theorem is referenced by:  xpeq12  4804  xpeq2i  4806  xpeq2d  4809  xpnz  5046  xpdisj2  5049  dmxpss  5053  rnxpid  5055  xpcan  5058  ovcross  5846  pmvalg  6011  xpcomeng  6054