Theorem axsep 5224

Description: Axiom scheme of separation ax-sep 5225 derived from the axiom scheme of replacement ax-rep 5206. The statement is identical to that of ax-sep 5225, and therefore shows that ax-sep 5225 is redundant when ax-rep 5206 is allowed. See ax-sep 5225 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5225 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 5223	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-rep 5206

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-eu 2573  df-rex 3065

This theorem is referenced by: (None)