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Mirrors > Home > MPE Home > Th. List > sbcel1v | Structured version Visualization version GIF version |
Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) Avoid ax-13 2374. (Revised by Wolf Lammen, 30-Apr-2023.) |
Ref | Expression |
---|---|
sbcel1v | ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3800 | . 2 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | elex 3498 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | dfsbcq2 3793 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑥 ∈ 𝐵)) | |
4 | eleq1 2826 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | clelsb1 2865 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵) | |
6 | 3, 4, 5 | vtoclbg 3556 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
7 | 1, 2, 6 | pm5.21nii 378 | 1 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 [wsb 2061 ∈ wcel 2105 Vcvv 3477 [wsbc 3790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-sbc 3791 |
This theorem is referenced by: tfinds2 7884 filuni 23908 gropeld 29064 grstructeld 29065 f1od2 32738 esum2dlem 34072 bnj110 34850 f1omptsnlem 37318 relowlpssretop 37346 rdgeqoa 37352 minregex 43523 cotrclrcl 43731 frege70 43922 frege72 43924 frege91 43943 sbcoreleleq 44532 onfrALTlem4 44540 sbcoreleleqVD 44856 onfrALTlem4VD 44883 rspesbcd 44935 |
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