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Mirrors > Home > MPE Home > Th. List > axrep3 | Structured version Visualization version GIF version |
Description: Axiom of Replacement slightly strengthened from axrep2 5184; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2381. (Revised by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
axrep3 | ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2145 | . . . 4 ⊢ Ⅎ𝑦∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) | |
2 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑥 | |
3 | nfv 1906 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝑤 | |
4 | nfa1 2146 | . . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦𝜑 | |
5 | 3, 4 | nfan 1891 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) |
6 | 5 | nfex 2334 | . . . . . 6 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) |
7 | 2, 6 | nfbi 1895 | . . . . 5 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) |
8 | 7 | nfal 2333 | . . . 4 ⊢ Ⅎ𝑦∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) |
9 | 1, 8 | nfim 1888 | . . 3 ⊢ Ⅎ𝑦(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))) |
10 | 9 | nfex 2334 | . 2 ⊢ Ⅎ𝑦∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))) |
11 | axreplem 5183 | . 2 ⊢ (𝑦 = 𝑤 → (∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))))) | |
12 | axrep2 5184 | . 2 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) | |
13 | 10, 11, 12 | chvarfv 2232 | 1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-rep 5181 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 |
This theorem is referenced by: axrep4 5186 |
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