Theorem axsep 5251

Description: Axiom scheme of separation ax-sep 5252 derived from the axiom scheme of replacement ax-rep 5235. The statement is identical to that of ax-sep 5252, and therefore shows that ax-sep 5252 is redundant when ax-rep 5235 is allowed. See ax-sep 5252 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5252 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 5250	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-rep 5235

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2567  df-eu 2597  df-rex 3091

This theorem is used by: (None)