Theorem axsep 5303

Description: Axiom scheme of separation ax-sep 5304 derived from the axiom scheme of replacement ax-rep 5290. The statement is identical to that of ax-sep 5304, and therefore shows that ax-sep 5304 is redundant when ax-rep 5290 is allowed. See ax-sep 5304 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5304 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 5302	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 394  ∀wal 1532  ∃wex 1774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-rep 5290

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-mo 2529  df-eu 2558  df-rex 3061

This theorem is referenced by: (None)

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