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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 2115). Version of clelsb1f 2909 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Rodolfo Medina, 28-Apr-2010.) Avoid ax-13 2372. (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 2891 | . . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 |
3 | 2 | sbco2v 2327 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴) |
4 | clelsb1 2861 | . . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
5 | 4 | sbbii 2080 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
6 | clelsb1 2861 | . 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 2068 ∈ wcel 2107 Ⅎwnfc 2884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-sb 2069 df-clel 2811 df-nfc 2886 |
This theorem is referenced by: rmo3f 3731 suppss2f 31863 fmptdF 31881 disjdsct 31924 esumpfinvalf 33074 |
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