Theorem wl-dfclab

Description: Rederive df-clab from wl-clabv . (Contributed by Wolf Lammen, 31-May-2023) (Proof modification is discouraged.)

	Ref	Expression
Assertion	wl-dfclab	$$⊢x\in \left\{y|\phi \right\}\leftrightarrow \left[x/y\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		dfclel	$$⊢x\in \left\{y|\phi \right\}\leftrightarrow \exists z\left(z=x\wedge z\in \left\{y|\phi \right\}\right)$$
2		wl-clabv	$$⊢z\in \left\{y|\phi \right\}\leftrightarrow \left[z/y\right]\phi$$
3		sbequ	$$⊢z=x\to \left(\left[z/y\right]\phi \leftrightarrow \left[x/y\right]\phi \right)$$
4	2 3	syl5bb	$$⊢z=x\to \left(z\in \left\{y|\phi \right\}\leftrightarrow \left[x/y\right]\phi \right)$$
5	4	pm5.32i	$$⊢\left(z=x\wedge z\in \left\{y|\phi \right\}\right)\leftrightarrow \left(z=x\wedge \left[x/y\right]\phi \right)$$
6	5	exbii	$$⊢\exists z\left(z=x\wedge z\in \left\{y|\phi \right\}\right)\leftrightarrow \exists z\left(z=x\wedge \left[x/y\right]\phi \right)$$
7		19.41v	$$⊢\exists z\left(z=x\wedge \left[x/y\right]\phi \right)\leftrightarrow \left(\exists zz=x\wedge \left[x/y\right]\phi \right)$$
8		ax6ev	$$⊢\exists zz=x$$
9	8	biantrur	$$⊢\left[x/y\right]\phi \leftrightarrow \left(\exists zz=x\wedge \left[x/y\right]\phi \right)$$
10	7 9	bitr4i	$$⊢\exists z\left(z=x\wedge \left[x/y\right]\phi \right)\leftrightarrow \left[x/y\right]\phi$$
11	1 6 10	3bitri	$$⊢x\in \left\{y|\phi \right\}\leftrightarrow \left[x/y\right]\phi$$