Theorem zfausclOLD

Description: Obsolete version of sepgi as of 14-Jul-2026. (Contributed by NM, 21-Jun-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sepgi.1	⊢ 𝐴 ∈ V
Assertion	zfausclOLD	⊢ ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sepgi.1	⊢ 𝐴 ∈ V
2		eleq2	⊢ ( 𝑧 = 𝐴 → ( 𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴 ) )
3	2	anbi1d	⊢ ( 𝑧 = 𝐴 → ( ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
4	3	bibi2d	⊢ ( 𝑧 = 𝐴 → ( ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ) )
5	4	albidv	⊢ ( 𝑧 = 𝐴 → ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝑧 ∧ 𝜑 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ) )
6	5	exbidv	⊢ ( 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	1 6 7	vtocl	⊢ ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )