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Mirrors > Home > MPE Home > Th. List > axsepg | Structured version Visualization version GIF version |
Description: A more general version of the axiom scheme of separation ax-sep 5300, where variable 𝑧 can also occur (in addition to 𝑥) in formula 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version is derived from the more restrictive ax-sep 5300 with no additional set theory axioms. Note that it was also derived from ax-rep 5286 but without ax-sep 5300 as axsepgfromrep 5298. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2172 and ax-13 2372 and shorten proof. (Revised by BJ, 6-Oct-2019.) |
Ref | Expression |
---|---|
axsepg | ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ2 2122 | . . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑧)) | |
2 | 1 | anbi1d 631 | . . . . 5 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))) |
3 | 2 | bibi2d 343 | . . . 4 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))) |
4 | 3 | albidv 1924 | . . 3 ⊢ (𝑤 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))) |
5 | 4 | exbidv 1925 | . 2 ⊢ (𝑤 = 𝑧 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))) |
6 | ax-sep 5300 | . 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) | |
7 | 5, 6 | chvarvv 2003 | 1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: (None) |
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