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Mirrors > Home > ILE Home > Th. List > clelsb3 | GIF version |
Description: Substitution applied to an atomic wff (class version of elsb3 1951). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
clelsb3 | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 | |
2 | 1 | sbco2 1938 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴) |
3 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴 | |
4 | eleq1 2202 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | sbie 1764 | . . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
6 | 5 | sbbii 1738 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
7 | nfv 1508 | . . 3 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴 | |
8 | eleq1 2202 | . . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
9 | 7, 8 | sbie 1764 | . 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
10 | 2, 6, 9 | 3bitr3i 209 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 [wsb 1735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-cleq 2132 df-clel 2135 |
This theorem is referenced by: hblem 2247 nfraldya 2469 nfrexdya 2470 cbvreu 2652 sbcel1v 2971 rmo3 3000 setindel 4453 elirr 4456 en2lp 4469 zfregfr 4488 tfi 4496 bdcriota 13081 |
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