Theorem xpeq1 4799

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

Assertion

Ref	Expression
xpeq1	⊢ (A = B → (A × C) = (B × 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 ∈ A ↔ x ∈ B))
2	1	anbi1d 685	. . 3 ⊢ (A = B → ((x ∈ A ∧ y ∈ C) ↔ (x ∈ B ∧ y ∈ C)))
3	2	opabbidv 4626	. 2 ⊢ (A = B → {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)} = {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)})
4		df-xp 4785	. 2 ⊢ (A × C) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)}
5		df-xp 4785	. 2 ⊢ (B × C) = {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)}
6	3, 4, 5	3eqtr4g 2410	1 ⊢ (A = B → (A × C) = (B × C))

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  xpeq1i  4805  xpeq1d  4808  dmxpid  4925  reseq2  4930  xpnz  5046  xpdisj1  5048  xpcan2  5059  ovcross  5846  pmvalg  6011  xpsneng  6051  xpcomeng  6054