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Mirrors > Home > ILE Home > Th. List > axsep2 | GIF version |
Description: A less restrictive version of the Separation Scheme ax-sep 3986, where variables 𝑥 and 𝑧 can both appear free in the wff 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version was derived from the more restrictive ax-sep 3986 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
axsep2 | ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2163 | . . . . . . 7 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑧)) | |
2 | 1 | anbi1d 456 | . . . . . 6 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑧 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑)))) |
3 | anabs5 543 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑧 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | |
4 | 2, 3 | syl6bb 195 | . . . . 5 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))) |
5 | 4 | bibi2d 231 | . . . 4 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑))) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))) |
6 | 5 | albidv 1763 | . . 3 ⊢ (𝑤 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑))) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))) |
7 | 6 | exbidv 1764 | . 2 ⊢ (𝑤 = 𝑧 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑))) ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))) |
8 | ax-sep 3986 | . 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑))) | |
9 | 7, 8 | chvarv 1872 | 1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1297 ∃wex 1436 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-ext 2082 ax-sep 3986 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-cleq 2093 df-clel 2096 |
This theorem is referenced by: (None) |
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