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Mirrors > Home > MPE Home > Th. List > abbi1dv | Structured version Visualization version GIF version |
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
Ref | Expression |
---|---|
abbi1dv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴)) |
Ref | Expression |
---|---|
abbi1dv | ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi1dv.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴)) | |
2 | 1 | bicomd 225 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
3 | 2 | abbi2dv 2952 | . 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
4 | 3 | eqcomd 2829 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {cab 2801 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 |
This theorem is referenced by: abidnf 3696 csbtt 3902 csbie2g 3925 csbvarg 4385 iinxsng 5012 predep 6176 enfin2i 9745 fin1a2lem11 9834 hashf1 13818 shftuz 14430 psrbaglefi 20154 vmappw 25695 hdmap1fval 38934 hdmapfval 38965 hgmapfval 39024 rabeqcda 39113 |
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