Theorem elabg

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of Quine p. 44. (Contributed by NM, 14-Apr-1995) Remove dependency on ax-13 . (Revised by Steven Nguyen, 23-Nov-2022)

	Ref	Expression
Hypothesis	elabg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	elabg	$$⊢A\in V\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elabg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		nfab1	$$⊢\underset{\_}{Ⅎ}x\left\{x|\phi \right\}$$
3	2	nfel2	$$⊢ℲxA\in \left\{x|\phi \right\}$$
4		nfv	$$⊢Ⅎx\psi$$
5	3 4	nfbi	$$⊢Ⅎx\left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$
6		eleq1	$$⊢x=A\to \left(x\in \left\{x|\phi \right\}\leftrightarrow A\in \left\{x|\phi \right\}\right)$$
7	6 1	bibi12d	$$⊢x=A\to \left(\left(x\in \left\{x|\phi \right\}\leftrightarrow \phi \right)\leftrightarrow \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)\right)$$
8		abid	$$⊢x\in \left\{x|\phi \right\}\leftrightarrow \phi$$
9	5 7 8	vtoclg1f	$$⊢A\in V\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$