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Mirrors > Home > ILE Home > Th. List > clelsb1 | GIF version |
Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2171). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
clelsb1 | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1539 | . . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 | |
2 | 1 | sbco2 1981 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴) |
3 | nfv 1539 | . . . 4 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴 | |
4 | eleq1 2256 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | sbie 1802 | . . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
6 | 5 | sbbii 1776 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
7 | nfv 1539 | . . 3 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴 | |
8 | eleq1 2256 | . . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
9 | 7, 8 | sbie 1802 | . 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
10 | 2, 6, 9 | 3bitr3i 210 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 [wsb 1773 ∈ wcel 2164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-cleq 2186 df-clel 2189 |
This theorem is referenced by: hblem 2301 eqabdv 2322 nfraldya 2529 nfrexdya 2530 cbvreu 2724 sbcel1v 3048 rmo3 3077 setindel 4570 elirr 4573 en2lp 4586 zfregfr 4606 tfi 4614 bdcriota 15375 |
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