Theorem csbeq2gVD

Description: Virtual deduction proof of csbeq2 . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbeq2 is csbeq2gVD without virtual deductions and was automatically derived from csbeq2gVD .

	Ref	Expression
Assertion	csbeq2gVD	$$⊢A\in V\to \left(\forall xB=C\to ⦋A/x⦌B=⦋A/x⦌C\right)$$

Proof

Step	Hyp	Ref	Expression
1		idn1	$$⊢\left(A\in V\to A\in V\right)$$
2		spsbc	$$⊢A\in V\to \left(\forall xB=C\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}B=C\right)$$
3	1 2	e1a	$$⊢\left(A\in V\to \left(\forall xB=C\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}B=C\right)\right)$$
4		sbceqg	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}B=C\leftrightarrow ⦋A/x⦌B=⦋A/x⦌C\right)$$
5	1 4	e1a	$$⊢\left(A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}B=C\leftrightarrow ⦋A/x⦌B=⦋A/x⦌C\right)\right)$$
6		imbi2	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}B=C\leftrightarrow ⦋A/x⦌B=⦋A/x⦌C\right)\to \left(\left(\forall xB=C\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}B=C\right)\leftrightarrow \left(\forall xB=C\to ⦋A/x⦌B=⦋A/x⦌C\right)\right)$$
7	6	biimpcd	$$⊢\left(\forall xB=C\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}B=C\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}B=C\leftrightarrow ⦋A/x⦌B=⦋A/x⦌C\right)\to \left(\forall xB=C\to ⦋A/x⦌B=⦋A/x⦌C\right)\right)$$
8	3 5 7	e11	$$⊢\left(A\in V\to \left(\forall xB=C\to ⦋A/x⦌B=⦋A/x⦌C\right)\right)$$
9	8	in1	$$⊢A\in V\to \left(\forall xB=C\to ⦋A/x⦌B=⦋A/x⦌C\right)$$