Theorem axsep 5301

Description: Axiom scheme of separation ax-sep 5302 derived from the axiom scheme of replacement ax-rep 5285. The statement is identical to that of ax-sep 5302, and therefore shows that ax-sep 5302 is redundant when ax-rep 5285 is allowed. See ax-sep 5302 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5302 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 5300	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-rep 5285

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-mo 2538  df-eu 2567  df-rex 3069

This theorem is referenced by: (None)