Theorem rabeq0w

Description: Condition for a restricted class abstraction to be empty. Version of rabeq0 using implicit substitution, which does not require ax-10 , ax-11 , ax-12 , but requires ax-8 . (Contributed by Gino Giotto, 30-Sep-2024)

	Ref	Expression
Hypothesis	rabeq0w.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rabeq0w	$$⊢\left\{x\in A|\phi \right\}=\varnothing \leftrightarrow \forall y\in A\neg \psi$$

Proof

Step	Hyp	Ref	Expression
1		rabeq0w.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
3	2 1	anbi12d	$$⊢x=y\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(y\in A\wedge \psi \right)\right)$$
4	3	ab0w	$$⊢\left\{x|\left(x\in A\wedge \phi \right)\right\}=\varnothing \leftrightarrow \forall y\neg \left(y\in A\wedge \psi \right)$$
5		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
6	5	eqeq1i	$$⊢\left\{x\in A|\phi \right\}=\varnothing \leftrightarrow \left\{x|\left(x\in A\wedge \phi \right)\right\}=\varnothing$$
7		raln	$$⊢\forall y\in A\neg \psi \leftrightarrow \forall y\neg \left(y\in A\wedge \psi \right)$$
8	4 6 7	3bitr4i	$$⊢\left\{x\in A|\phi \right\}=\varnothing \leftrightarrow \forall y\in A\neg \psi$$