Theorem abbiOLD

Description: Obsolete proof of abbi as of 7-Jan-2024. (Contributed by NM, 25-Nov-2013) (Revised by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 16-Nov-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	abbiOLD	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left\{x|\phi \right\}=\left\{x|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		hbab1	$$⊢y\in \left\{x|\phi \right\}\to \forall xy\in \left\{x|\phi \right\}$$
2		hbab1	$$⊢y\in \left\{x|\psi \right\}\to \forall xy\in \left\{x|\psi \right\}$$
3	1 2	cleqh	$$⊢\left\{x|\phi \right\}=\left\{x|\psi \right\}\leftrightarrow \forall x\left(x\in \left\{x|\phi \right\}\leftrightarrow x\in \left\{x|\psi \right\}\right)$$
4		abid	$$⊢x\in \left\{x|\phi \right\}\leftrightarrow \phi$$
5		abid	$$⊢x\in \left\{x|\psi \right\}\leftrightarrow \psi$$
6	4 5	bibi12i	$$⊢\left(x\in \left\{x|\phi \right\}\leftrightarrow x\in \left\{x|\psi \right\}\right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)$$
7	6	albii	$$⊢\forall x\left(x\in \left\{x|\phi \right\}\leftrightarrow x\in \left\{x|\psi \right\}\right)\leftrightarrow \forall x\left(\phi \leftrightarrow \psi \right)$$
8	3 7	bitr2i	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left\{x|\phi \right\}=\left\{x|\psi \right\}$$