![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > clelsb3f | Structured version Visualization version GIF version |
Description: Substitution applied to an atomic wff (class version of elsb3 2583). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) |
Ref | Expression |
---|---|
clelsb3f.1 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
clelsb3f | ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelsb3f.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
2 | 1 | nfcri 2907 | . . 3 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴 |
3 | 2 | sbco2 2562 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴) |
4 | nfv 1995 | . . . 4 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴 | |
5 | eleq1w 2833 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
6 | 4, 5 | sbie 2555 | . . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
7 | 6 | sbbii 2056 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴) |
8 | nfv 1995 | . . 3 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴 | |
9 | eleq1w 2833 | . . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
10 | 8, 9 | sbie 2555 | . 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
11 | 3, 7, 10 | 3bitr3i 290 | 1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 [wsb 2049 ∈ wcel 2145 Ⅎwnfc 2900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clel 2767 df-nfc 2902 |
This theorem is referenced by: rmo3f 29674 suppss2f 29779 fmptdF 29796 disjdsct 29820 esumpfinvalf 30478 |
Copyright terms: Public domain | W3C validator |