Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > clelsb3fw | Structured version Visualization version GIF version |
Description: Substitution applied to an atomic wff (class version of elsb3 2121). Version of clelsb3f 2981 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
clelsb3fw.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
clelsb3fw | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelsb3fw.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcri 2970 | . . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 |
3 | 2 | sbco2v 2351 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴) |
4 | clelsb3 2939 | . . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
5 | 4 | sbbii 2080 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
6 | clelsb3 2939 | . 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
7 | 3, 5, 6 | 3bitr3i 303 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 [wsb 2068 ∈ wcel 2113 Ⅎwnfc 2960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-sb 2069 df-clel 2892 df-nfc 2962 |
This theorem is referenced by: rmo3f 3721 suppss2f 30384 fmptdF 30401 disjdsct 30438 esumpfinvalf 31356 |
Copyright terms: Public domain | W3C validator |