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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 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Axiom ax-sep
StepHypRef Expression
1 vx . . . . 5 setvar 𝑥
2 vy . . . . 5 setvar 𝑦
31, 2wel 2031 . . . 4 wff 𝑥𝑦
4 vz . . . . . 6 setvar 𝑧
51, 4wel 2031 . . . . 5 wff 𝑥𝑧
6 wph . . . . 5 wff 𝜑
75, 6wa 383 . . . 4 wff (𝑥𝑧𝜑)
83, 7wb 196 . . 3 wff (𝑥𝑦 ↔ (𝑥𝑧𝜑))
98, 1wal 1521 . 2 wff 𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
109, 2wex 1744 1 wff 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
This axiom is referenced by:  axsep2  4815  zfauscl  4816  bm1.3ii  4817  ax6vsep  4818  axnul  4821  nalset  4828  bj-nalset  32919  bj-axsep2  33046
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