Theorem bnj1441

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) Add disjoint variable condition to avoid ax-13 . See bnj1441g for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1441.1	$$⊢x\in A\to \forall yx\in A$$
	bnj1441.2	$$⊢\phi \to \forall y\phi$$
Assertion	bnj1441	$$⊢z\in \left\{x\in A|\phi \right\}\to \forall yz\in \left\{x\in A|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		bnj1441.1	$$⊢x\in A\to \forall yx\in A$$
2		bnj1441.2	$$⊢\phi \to \forall y\phi$$
3		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
4	1 2	hban	$$⊢\left(x\in A\wedge \phi \right)\to \forall y\left(x\in A\wedge \phi \right)$$
5	4	hbab	$$⊢z\in \left\{x|\left(x\in A\wedge \phi \right)\right\}\to \forall yz\in \left\{x|\left(x\in A\wedge \phi \right)\right\}$$
6	3 5	hbxfreq	$$⊢z\in \left\{x\in A|\phi \right\}\to \forall yz\in \left\{x\in A|\phi \right\}$$