Theorem nfcsbd

Description: Deduction version of nfcsb . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 21-Nov-2005) (Revised by Mario Carneiro, 12-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfcsbd.1	$$⊢Ⅎy\phi$$
	nfcsbd.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
	nfcsbd.3	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
Assertion	nfcsbd	$$⊢\phi \to \underset{\_}{Ⅎ}x⦋A/y⦌B$$

Proof

Step	Hyp	Ref	Expression
1		nfcsbd.1	$$⊢Ⅎy\phi$$
2		nfcsbd.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
3		nfcsbd.3	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
4		df-csb	$$⊢⦋A/y⦌B=\left\{z|\underset{\dot{}}{[}A/y\underset{\dot{}}{]}z\in B\right\}$$
5		nfv	$$⊢Ⅎz\phi$$
6	3	nfcrd	$$⊢\phi \to Ⅎxz\in B$$
7	1 2 6	nfsbcd	$$⊢\phi \to Ⅎx\underset{\dot{}}{[}A/y\underset{\dot{}}{]}z\in B$$
8	5 7	nfabd	$$⊢\phi \to \underset{\_}{Ⅎ}x\left\{z|\underset{\dot{}}{[}A/y\underset{\dot{}}{]}z\in B\right\}$$
9	4 8	nfcxfrd	$$⊢\phi \to \underset{\_}{Ⅎ}x⦋A/y⦌B$$