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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-zfauscl | Structured version Visualization version GIF version |
Description: General version of zfauscl 5207.
Remark: the comment in zfauscl 5207 is misleading: the essential use of ax-ext 2795 is the one via eleq2 2903 and not the one via vtocl 3561, since the latter can be proved without ax-ext 2795 (see bj-vtoclg 34238). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-zfauscl | ⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2903 | . . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴)) | |
2 | 1 | anbi1d 631 | . . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
3 | 2 | bibi2d 345 | . . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
4 | 3 | biimpd 231 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
5 | 4 | alimdv 1917 | . . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
6 | 5 | eximdv 1918 | . 2 ⊢ (𝑧 = 𝐴 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
7 | ax-sep 5205 | . 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | |
8 | 6, 7 | bj-vtoclg 34238 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 ax-sep 5205 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2816 df-clel 2895 |
This theorem is referenced by: (None) |
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