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Mirrors > Home > MPE Home > Th. List > axsep | Structured version Visualization version GIF version |
Description: Axiom scheme of separation ax-sep 5167 derived from the axiom scheme of replacement ax-rep 5154. The statement is identical to that of ax-sep 5167, and therefore shows that ax-sep 5167 is redundant when ax-rep 5154 is allowed. See ax-sep 5167 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5167 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
axsep | ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axsepgfromrep 5165 | 1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∀wal 1540 ∃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 1975 ax-7 2020 ax-8 2116 ax-rep 5154 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-mo 2540 df-eu 2570 df-rex 3059 |
This theorem is referenced by: (None) |
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