Theorem bj-sepg

Description: Version of sepg which does not require df-clab , thanks to the use of bj-vtoclg (and ultimately, to the use of elissetv instead of elisset ). This axiom save is not very important, since this theorem uses df-cleq and df-clel . This theorem is a remnant of a previous state of set.mm where the axiom saving was larger. (Contributed by BJ, 2-Jul-2022) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-sepg	⊢ ( 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 ) ) )