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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-clelsb3 | Structured version Visualization version GIF version |
Description: Remove dependency on ax-ext 2590 (and df-cleq 2603) from clelsb3 2716. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-clelsb3 | ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 | |
2 | 1 | sbco2 2403 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ [𝑥 / 𝑧]𝑧 ∈ 𝐴) |
3 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐴 | |
4 | bj-eleq1w 32040 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
5 | 3, 4 | sbie 2396 | . . 3 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
6 | 5 | sbbii 1874 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴) |
7 | nfv 1830 | . . 3 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴 | |
8 | bj-eleq1w 32040 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
9 | 7, 8 | sbie 2396 | . 2 ⊢ ([𝑥 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
10 | 2, 6, 9 | 3bitr3i 289 | 1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 [wsb 1867 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clel 2606 |
This theorem is referenced by: bj-hblem 32043 |
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