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Mirrors > Home > MPE Home > Th. List > axsepgfromrep | Structured version Visualization version GIF version |
Description: A more general version axsepg 5168 of the axiom scheme of separation ax-sep 5167 derived from the axiom scheme of replacement ax-rep 5154 (and first-order logic). The extra generality consists in the absence of a disjoint variable condition on 𝑧, 𝜑 (that is, variable 𝑧 may occur in formula 𝜑). See linked statements for more information. (Contributed by NM, 11-Sep-2006.) Remove dependencies on ax-9 2121 to ax-13 2379. (Revised by SN, 25-Sep-2023.) Use ax-sep 5167 instead (or axsepg 5168 if the extra generality is needed). (New usage is discouraged.) |
Ref | Expression |
---|---|
axsepgfromrep | ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axrep6 5161 | . . 3 ⊢ (∀𝑤∃*𝑥(𝑤 = 𝑥 ∧ 𝜑) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑧 (𝑤 = 𝑥 ∧ 𝜑))) | |
2 | euequ 2658 | . . . . 5 ⊢ ∃!𝑥 𝑥 = 𝑤 | |
3 | 2 | eumoi 2639 | . . . 4 ⊢ ∃*𝑥 𝑥 = 𝑤 |
4 | equcomi 2024 | . . . . . 6 ⊢ (𝑤 = 𝑥 → 𝑥 = 𝑤) | |
5 | 4 | adantr 484 | . . . . 5 ⊢ ((𝑤 = 𝑥 ∧ 𝜑) → 𝑥 = 𝑤) |
6 | 5 | moimi 2603 | . . . 4 ⊢ (∃*𝑥 𝑥 = 𝑤 → ∃*𝑥(𝑤 = 𝑥 ∧ 𝜑)) |
7 | 3, 6 | ax-mp 5 | . . 3 ⊢ ∃*𝑥(𝑤 = 𝑥 ∧ 𝜑) |
8 | 1, 7 | mpg 1799 | . 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑧 (𝑤 = 𝑥 ∧ 𝜑)) |
9 | df-rex 3112 | . . . . . 6 ⊢ (∃𝑤 ∈ 𝑧 (𝑤 = 𝑥 ∧ 𝜑) ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ (𝑤 = 𝑥 ∧ 𝜑))) | |
10 | an12 644 | . . . . . . 7 ⊢ ((𝑤 = 𝑥 ∧ (𝑤 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑤 ∈ 𝑧 ∧ (𝑤 = 𝑥 ∧ 𝜑))) | |
11 | 10 | exbii 1849 | . . . . . 6 ⊢ (∃𝑤(𝑤 = 𝑥 ∧ (𝑤 ∈ 𝑧 ∧ 𝜑)) ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ (𝑤 = 𝑥 ∧ 𝜑))) |
12 | elequ1 2118 | . . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧)) | |
13 | 12 | anbi1d 632 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))) |
14 | 13 | equsexvw 2011 | . . . . . 6 ⊢ (∃𝑤(𝑤 = 𝑥 ∧ (𝑤 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
15 | 9, 11, 14 | 3bitr2i 302 | . . . . 5 ⊢ (∃𝑤 ∈ 𝑧 (𝑤 = 𝑥 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
16 | 15 | bibi2i 341 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑧 (𝑤 = 𝑥 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))) |
17 | 16 | albii 1821 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑧 (𝑤 = 𝑥 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))) |
18 | 17 | exbii 1849 | . 2 ⊢ (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑧 (𝑤 = 𝑥 ∧ 𝜑)) ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))) |
19 | 8, 18 | mpbi 233 | 1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃wex 1781 ∃*wmo 2596 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-rep 5154 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-mo 2598 df-eu 2629 df-rex 3112 |
This theorem is referenced by: axsep 5166 |
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