Theorem bj-zfauscl

Description: General version of zfauscl .

Remark: the comment in zfauscl is misleading: the essential use of ax-ext is the one via eleq2 and not the one via vtocl , since the latter can be proved without ax-ext (see bj-vtoclg ).

(Contributed by BJ, 2-Jul-2022) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-zfauscl	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝑧 = 𝐴 → ( 𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴 ) )
2	1	anbi1d	⊢ ( 𝑧 = 𝐴 → ( ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
3	2	bibi2d	⊢ ( 𝑧 = 𝐴 → ( ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ) )
4	3	biimpd	⊢ ( 𝑧 = 𝐴 → ( ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) ) → ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ) )
5	4	alimdv	⊢ ( 𝑧 = 𝐴 → ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) ) → ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ) )
6	5	eximdv	⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) ) → ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ) )
7		ax-sep	⊢ ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) )
8	6 7	bj-vtoclg	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )