![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > clelsb1f | Structured version Visualization version GIF version |
Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2113). Usage of this theorem is discouraged because it depends on ax-13 2370. See clelsb1fw 2906 not requiring ax-13 2370, but extra disjoint variables. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) (Proof shortened by Wolf Lammen, 7-May-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
clelsb1f.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
clelsb1f | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelsb1f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcri 2889 | . . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 |
3 | 2 | sbco2 2509 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴) |
4 | clelsb1 2859 | . . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
5 | 4 | sbbii 2078 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
6 | clelsb1 2859 | . 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
7 | 3, 5, 6 | 3bitr3i 301 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2066 ∈ wcel 2105 Ⅎwnfc 2882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-10 2136 ax-11 2153 ax-12 2170 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clel 2809 df-nfc 2884 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |