Theorem altxpeq2

Description: Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012)

	Ref	Expression
Assertion	altxpeq2	$$⊢A=B\to \left(C\mathrm{\times \times }A\right)=\left(C\mathrm{\times \times }B\right)$$

Proof

Step	Hyp	Ref	Expression
1		rexeq	$$⊢A=B\to \left(\exists y\in Az=⟪x,y⟫\leftrightarrow \exists y\in Bz=⟪x,y⟫\right)$$
2	1	rexbidv	$$⊢A=B\to \left(\exists x\in C\exists y\in Az=⟪x,y⟫\leftrightarrow \exists x\in C\exists y\in Bz=⟪x,y⟫\right)$$
3	2	abbidv	$$⊢A=B\to \left\{z|\exists x\in C\exists y\in Az=⟪x,y⟫\right\}=\left\{z|\exists x\in C\exists y\in Bz=⟪x,y⟫\right\}$$
4		df-altxp	$$⊢\left(C\mathrm{\times \times }A\right)=\left\{z|\exists x\in C\exists y\in Az=⟪x,y⟫\right\}$$
5		df-altxp	$$⊢\left(C\mathrm{\times \times }B\right)=\left\{z|\exists x\in C\exists y\in Bz=⟪x,y⟫\right\}$$
6	3 4 5	3eqtr4g	$$⊢A=B\to \left(C\mathrm{\times \times }A\right)=\left(C\mathrm{\times \times }B\right)$$