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) Avoid ax-13 . (Revised by SN, 23-Nov-2022) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 5-Oct-2024)

	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		elab6g	$$⊢A\in V\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \forall x\left(x=A\to \phi \right)\right)$$
3	1	pm5.74i	$$⊢\left(x=A\to \phi \right)\leftrightarrow \left(x=A\to \psi \right)$$
4	3	albii	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \forall x\left(x=A\to \psi \right)$$
5		19.23v	$$⊢\forall x\left(x=A\to \psi \right)\leftrightarrow \left(\exists xx=A\to \psi \right)$$
6	4 5	bitri	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \left(\exists xx=A\to \psi \right)$$
7		elisset	$$⊢A\in V\to \exists xx=A$$
8		pm5.5	$$⊢\exists xx=A\to \left(\left(\exists xx=A\to \psi \right)\leftrightarrow \psi \right)$$
9	7 8	syl	$$⊢A\in V\to \left(\left(\exists xx=A\to \psi \right)\leftrightarrow \psi \right)$$
10	6 9	bitrid	$$⊢A\in V\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$
11	2 10	bitrd	$$⊢A\in V\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$