Theorem wl-clelsb3df

Description: Deduction version of clelsb3f . (Contributed by Wolf Lammen, 29-May-2023)

	Ref	Expression
Hypotheses	clelsb3df.1	$$⊢Ⅎy\phi$$
	clelsb3df.2	$$⊢\phi \to \underset{\_}{Ⅎ}yA$$
Assertion	wl-clelsb3df	$$⊢\phi \to \left(\left[x/y\right]y\in A\leftrightarrow x\in A\right)$$

Proof

Step	Hyp	Ref	Expression
1		clelsb3df.1	$$⊢Ⅎy\phi$$
2		clelsb3df.2	$$⊢\phi \to \underset{\_}{Ⅎ}yA$$
3		nfv	$$⊢Ⅎw\phi$$
4	2	nfcrd	$$⊢\phi \to Ⅎyw\in A$$
5	3 1 4	sbco2d	$$⊢\phi \to \left(\left[x/y\right]\left[y/w\right]w\in A\leftrightarrow \left[x/w\right]w\in A\right)$$
6		clelsb3	$$⊢\left[y/w\right]w\in A\leftrightarrow y\in A$$
7	6	sbbii	$$⊢\left[x/y\right]\left[y/w\right]w\in A\leftrightarrow \left[x/y\right]y\in A$$
8		clelsb3	$$⊢\left[x/w\right]w\in A\leftrightarrow x\in A$$
9	5 7 8	3bitr3g	$$⊢\phi \to \left(\left[x/y\right]y\in A\leftrightarrow x\in A\right)$$