Theorem xpeq1

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

	Ref	Expression
Assertion	xpeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
2	1	anbi1d	⊢ ( 𝐴 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )
3	2	opabbidv	⊢ ( 𝐴 = 𝐵 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) } )
4		df-xp	⊢ ( 𝐴 × 𝐶 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) }
5		df-xp	⊢ ( 𝐵 × 𝐶 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) }
6	3 4 5	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐶 ) )