Theorem axsep 5166

Description: Axiom scheme of separation ax-sep 5167 derived from the axiom scheme of replacement ax-rep 5154. The statement is identical to that of ax-sep 5167, and therefore shows that ax-sep 5167 is redundant when ax-rep 5154 is allowed. See ax-sep 5167 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5167 instead. (New usage is discouraged.)

Assertion

Ref	Expression
axsep	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem axsep

Step	Hyp	Ref	Expression
1		axsepgfromrep 5165	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-rep 5154

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2598  df-eu 2629  df-rex 3112

This theorem is referenced by: (None)