Axiom ax-sep 4814

Description: The Axiom of Separation of ZF set theory. See axsep 4813 for more information. It was derived as axsep 4813 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 11-Sep-2006.)

Assertion

Ref	Expression
ax-sep	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Axiom ax-sep

Step	Hyp	Ref	Expression
1		vx	. . . . 5 setvar 𝑥
2		vy	. . . . 5 setvar 𝑦
3	1, 2	wel 2031	. . . 4 wff 𝑥 ∈ 𝑦
4		vz	. . . . . 6 setvar 𝑧
5	1, 4	wel 2031	. . . . 5 wff 𝑥 ∈ 𝑧
6		wph	. . . . 5 wff 𝜑
7	5, 6	wa 383	. . . 4 wff (𝑥 ∈ 𝑧 ∧ 𝜑)
8	3, 7	wb 196	. . 3 wff (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
9	8, 1	wal 1521	. 2 wff ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
10	9, 2	wex 1744	1 wff ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

This axiom is referenced by:  axsep2  4815  zfauscl  4816  bm1.3ii  4817  ax6vsep  4818  axnul  4821  nalset  4828  bj-nalset  32919  bj-axsep2  33046