Theorem elabgt

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg .) (Contributed by NM, 7-Nov-2005) (Proof shortened by Andrew Salmon, 8-Jun-2011) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024)

	Ref	Expression
Assertion	elabgt	$$⊢\left(A\in B\wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elab6g	$$⊢A\in B\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \forall x\left(x=A\to \phi \right)\right)$$
2	1	adantr	$$⊢\left(A\in B\wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \forall x\left(x=A\to \phi \right)\right)$$
3		elisset	$$⊢A\in B\to \exists xx=A$$
4		biimp	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\phi \to \psi \right)$$
5	4	imim3i	$$⊢\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\left(x=A\to \phi \right)\to \left(x=A\to \psi \right)\right)$$
6	5	al2imi	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\forall x\left(x=A\to \phi \right)\to \forall x\left(x=A\to \psi \right)\right)$$
7		19.23v	$$⊢\forall x\left(x=A\to \psi \right)\leftrightarrow \left(\exists xx=A\to \psi \right)$$
8	6 7	syl6ib	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\forall x\left(x=A\to \phi \right)\to \left(\exists xx=A\to \psi \right)\right)$$
9	8	com3r	$$⊢\exists xx=A\to \left(\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\forall x\left(x=A\to \phi \right)\to \psi \right)\right)$$
10		biimpr	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)$$
11	10	imim2i	$$⊢\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(x=A\to \left(\psi \to \phi \right)\right)$$
12	11	alimi	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \forall x\left(x=A\to \left(\psi \to \phi \right)\right)$$
13		bi2.04	$$⊢\left(x=A\to \left(\psi \to \phi \right)\right)\leftrightarrow \left(\psi \to \left(x=A\to \phi \right)\right)$$
14	13	albii	$$⊢\forall x\left(x=A\to \left(\psi \to \phi \right)\right)\leftrightarrow \forall x\left(\psi \to \left(x=A\to \phi \right)\right)$$
15		19.21v	$$⊢\forall x\left(\psi \to \left(x=A\to \phi \right)\right)\leftrightarrow \left(\psi \to \forall x\left(x=A\to \phi \right)\right)$$
16	14 15	sylbb	$$⊢\forall x\left(x=A\to \left(\psi \to \phi \right)\right)\to \left(\psi \to \forall x\left(x=A\to \phi \right)\right)$$
17	12 16	syl	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\psi \to \forall x\left(x=A\to \phi \right)\right)$$
18	17	a1i	$$⊢\exists xx=A\to \left(\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\psi \to \forall x\left(x=A\to \phi \right)\right)\right)$$
19	9 18	impbidd	$$⊢\exists xx=A\to \left(\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)\right)$$
20	3 19	syl	$$⊢A\in B\to \left(\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)\right)$$
21	20	imp	$$⊢\left(A\in B\wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$
22	2 21	bitrd	$$⊢\left(A\in B\wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$