Theorem bj-projeq

Description: Substitution property for Proj . (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-projeq	$$⊢A=C\to \left(B=D\to \left(AProjB\right)=\left(CProjD\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		simpr	$$⊢\left(A=C\wedge B=D\right)\to B=D$$
2		simpl	$$⊢\left(A=C\wedge B=D\right)\to A=C$$
3	2	sneqd	$$⊢\left(A=C\wedge B=D\right)\to \left\{A\right\}=\left\{C\right\}$$
4	1 3	imaeq12d	$$⊢\left(A=C\wedge B=D\right)\to B\left[\left\{A\right\}\right]=D\left[\left\{C\right\}\right]$$
5	4	eleq2d	$$⊢\left(A=C\wedge B=D\right)\to \left(\left\{x\right\}\in B\left[\left\{A\right\}\right]\leftrightarrow \left\{x\right\}\in D\left[\left\{C\right\}\right]\right)$$
6	5	abbidv	$$⊢\left(A=C\wedge B=D\right)\to \left\{x|\left\{x\right\}\in B\left[\left\{A\right\}\right]\right\}=\left\{x|\left\{x\right\}\in D\left[\left\{C\right\}\right]\right\}$$
7		df-bj-proj	$$⊢\left(AProjB\right)=\left\{x|\left\{x\right\}\in B\left[\left\{A\right\}\right]\right\}$$
8		df-bj-proj	$$⊢\left(CProjD\right)=\left\{x|\left\{x\right\}\in D\left[\left\{C\right\}\right]\right\}$$
9	6 7 8	3eqtr4g	$$⊢\left(A=C\wedge B=D\right)\to \left(AProjB\right)=\left(CProjD\right)$$
10	9	ex	$$⊢A=C\to \left(B=D\to \left(AProjB\right)=\left(CProjD\right)\right)$$