Theorem elabd2

Description: Membership in a class abstraction, using implicit substitution. Deduction version of elab . (Contributed by Gino Giotto, 12-Oct-2024) (Revised by BJ, 16-Oct-2024)

	Ref	Expression
Hypotheses	elabd2.ex	$$⊢\phi \to A\in V$$
	elabd2.eq	$$⊢\phi \to B=\left\{x|\psi \right\}$$
	elabd2.is	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	elabd2	$$⊢\phi \to \left(A\in B\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elabd2.ex	$$⊢\phi \to A\in V$$
2		elabd2.eq	$$⊢\phi \to B=\left\{x|\psi \right\}$$
3		elabd2.is	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
4	2	eleq2d	$$⊢\phi \to \left(A\in B\leftrightarrow A\in \left\{x|\psi \right\}\right)$$
5		elab6g	$$⊢A\in V\to \left(A\in \left\{x|\psi \right\}\leftrightarrow \forall x\left(x=A\to \psi \right)\right)$$
6	4 5	sylan9bb	$$⊢\left(\phi \wedge A\in V\right)\to \left(A\in B\leftrightarrow \forall x\left(x=A\to \psi \right)\right)$$
7		elisset	$$⊢A\in V\to \exists xx=A$$
8	3	pm5.74da	$$⊢\phi \to \left(\left(x=A\to \psi \right)\leftrightarrow \left(x=A\to \chi \right)\right)$$
9	8	albidv	$$⊢\phi \to \left(\forall x\left(x=A\to \psi \right)\leftrightarrow \forall x\left(x=A\to \chi \right)\right)$$
10		19.23v	$$⊢\forall x\left(x=A\to \chi \right)\leftrightarrow \left(\exists xx=A\to \chi \right)$$
11	9 10	bitrdi	$$⊢\phi \to \left(\forall x\left(x=A\to \psi \right)\leftrightarrow \left(\exists xx=A\to \chi \right)\right)$$
12		pm5.5	$$⊢\exists xx=A\to \left(\left(\exists xx=A\to \chi \right)\leftrightarrow \chi \right)$$
13	11 12	sylan9bb	$$⊢\left(\phi \wedge \exists xx=A\right)\to \left(\forall x\left(x=A\to \psi \right)\leftrightarrow \chi \right)$$
14	7 13	sylan2	$$⊢\left(\phi \wedge A\in V\right)\to \left(\forall x\left(x=A\to \psi \right)\leftrightarrow \chi \right)$$
15	6 14	bitrd	$$⊢\left(\phi \wedge A\in V\right)\to \left(A\in B\leftrightarrow \chi \right)$$
16	1 15	mpdan	$$⊢\phi \to \left(A\in B\leftrightarrow \chi \right)$$