Theorem xpeq1

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

	Ref	Expression
Assertion	xpeq1	$$⊢A=B\to A\times C=B\times C$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$
2	1	anbi1d	$$⊢A=B\to \left(\left(x\in A\wedge y\in C\right)\leftrightarrow \left(x\in B\wedge y\in C\right)\right)$$
3	2	opabbidv	$$⊢A=B\to \left\{⟨x,y⟩|\left(x\in A\wedge y\in C\right)\right\}=\left\{⟨x,y⟩|\left(x\in B\wedge y\in C\right)\right\}$$
4		df-xp	$$⊢A\times C=\left\{⟨x,y⟩|\left(x\in A\wedge y\in C\right)\right\}$$
5		df-xp	$$⊢B\times C=\left\{⟨x,y⟩|\left(x\in B\wedge y\in C\right)\right\}$$
6	3 4 5	3eqtr4g	$$⊢A=B\to A\times C=B\times C$$