![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2371. (Revised by Wolf Lammen, 30-Apr-2023.) |
Ref | Expression |
---|---|
sbcel1v | ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3787 | . 2 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | elex 3492 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | dfsbcq2 3780 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑥 ∈ 𝐵)) | |
4 | eleq1 2821 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | clelsb1 2860 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵) | |
6 | 3, 4, 5 | vtoclbg 3559 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
7 | 1, 2, 6 | pm5.21nii 379 | 1 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2067 ∈ wcel 2106 Vcvv 3474 [wsbc 3777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-sbc 3778 |
This theorem is referenced by: tfinds2 7855 filuni 23609 gropeld 28548 grstructeld 28549 f1od2 32201 esum2dlem 33376 bnj110 34155 f1omptsnlem 36520 relowlpssretop 36548 rdgeqoa 36554 minregex 42587 cotrclrcl 42795 frege70 42986 frege72 42988 frege91 43007 sbcoreleleq 43598 onfrALTlem4 43606 sbcoreleleqVD 43922 onfrALTlem4VD 43949 |
Copyright terms: Public domain | W3C validator |