Theorem xpeq2

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

	Ref	Expression
Assertion	xpeq2	$$⊢A=B\to C\times A=C\times B$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢A=B\to \left(y\in A\leftrightarrow y\in B\right)$$
2	1	anbi2d	$$⊢A=B\to \left(\left(x\in C\wedge y\in A\right)\leftrightarrow \left(x\in C\wedge y\in B\right)\right)$$
3	2	opabbidv	$$⊢A=B\to \left\{⟨x,y⟩|\left(x\in C\wedge y\in A\right)\right\}=\left\{⟨x,y⟩|\left(x\in C\wedge y\in B\right)\right\}$$
4		df-xp	$$⊢C\times A=\left\{⟨x,y⟩|\left(x\in C\wedge y\in A\right)\right\}$$
5		df-xp	$$⊢C\times B=\left\{⟨x,y⟩|\left(x\in C\wedge y\in B\right)\right\}$$
6	3 4 5	3eqtr4g	$$⊢A=B\to C\times A=C\times B$$