Theorem abbib

Description: Equal class abstractions require equivalent formulas, and conversely. (Contributed by NM, 25-Nov-2013) (Revised by Mario Carneiro, 11-Aug-2016) Remove dependency on ax-8 and df-clel (by avoiding use of cleqh ). (Revised by BJ, 23-Jun-2019) Definitial form. (Revised by Wolf Lammen, 23-Feb-2025)

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

Proof

Step	Hyp	Ref	Expression
1		dfcleq	$$⊢\left\{x|\phi \right\}=\left\{x|\psi \right\}\leftrightarrow \forall y\left(y\in \left\{x|\phi \right\}\leftrightarrow y\in \left\{x|\psi \right\}\right)$$
2		nfsab1	$$⊢Ⅎxy\in \left\{x|\phi \right\}$$
3		nfsab1	$$⊢Ⅎxy\in \left\{x|\psi \right\}$$
4	2 3	nfbi	$$⊢Ⅎx\left(y\in \left\{x|\phi \right\}\leftrightarrow y\in \left\{x|\psi \right\}\right)$$
5		nfv	$$⊢Ⅎy\left(\phi \leftrightarrow \psi \right)$$
6		df-clab	$$⊢y\in \left\{x|\phi \right\}\leftrightarrow \left[y/x\right]\phi$$
7		sbequ12r	$$⊢y=x\to \left(\left[y/x\right]\phi \leftrightarrow \phi \right)$$
8	6 7	bitrid	$$⊢y=x\to \left(y\in \left\{x|\phi \right\}\leftrightarrow \phi \right)$$
9		df-clab	$$⊢y\in \left\{x|\psi \right\}\leftrightarrow \left[y/x\right]\psi$$
10		sbequ12r	$$⊢y=x\to \left(\left[y/x\right]\psi \leftrightarrow \psi \right)$$
11	9 10	bitrid	$$⊢y=x\to \left(y\in \left\{x|\psi \right\}\leftrightarrow \psi \right)$$
12	8 11	bibi12d	$$⊢y=x\to \left(\left(y\in \left\{x|\phi \right\}\leftrightarrow y\in \left\{x|\psi \right\}\right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)\right)$$
13	4 5 12	cbvalv1	$$⊢\forall y\left(y\in \left\{x|\phi \right\}\leftrightarrow y\in \left\{x|\psi \right\}\right)\leftrightarrow \forall x\left(\phi \leftrightarrow \psi \right)$$
14	1 13	bitri	$$⊢\left\{x|\phi \right\}=\left\{x|\psi \right\}\leftrightarrow \forall x\left(\phi \leftrightarrow \psi \right)$$