Theorem axsep 5217

Description: Axiom scheme of separation ax-sep 5218 derived from the axiom scheme of replacement ax-rep 5205. The statement is identical to that of ax-sep 5218, and therefore shows that ax-sep 5218 is redundant when ax-rep 5205 is allowed. See ax-sep 5218 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5218 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 5216	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-rep 5205

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-eu 2569  df-rex 3069

This theorem is referenced by: (None)