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Mirrors > Home > MPE Home > Th. List > clelsb1fw | Structured version Visualization version GIF version |
Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2112). Version of clelsb1f 2906 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Rodolfo Medina, 28-Apr-2010.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
clelsb1fw.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
clelsb1fw | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelsb1fw.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcri 2888 | . . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 |
3 | 2 | sbco2v 2324 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴) |
4 | clelsb1 2858 | . . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
5 | 4 | sbbii 2077 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
6 | clelsb1 2858 | . 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
7 | 3, 5, 6 | 3bitr3i 300 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2065 ∈ wcel 2104 Ⅎwnfc 2881 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-10 2135 ax-11 2152 ax-12 2169 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-nf 1784 df-sb 2066 df-clel 2808 df-nfc 2883 |
This theorem is referenced by: rmo3f 3729 suppss2f 32130 fmptdF 32148 disjdsct 32191 esumpfinvalf 33372 |
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