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Mirrors > Home > MPE Home > Th. List > eqabdv | Structured version Visualization version GIF version |
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) Avoid ax-11 2155. (Revised by Wolf Lammen, 6-May-2023.) |
Ref | Expression |
---|---|
eqabdv.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Ref | Expression |
---|---|
eqabdv | ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqabdv.1 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) | |
2 | 1 | sbbidv 2077 | . . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝜓)) |
3 | clelsb1 2866 | . . . 4 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
4 | 3 | bicomi 224 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
5 | df-clab 2713 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
6 | 2, 4, 5 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ {𝑥 ∣ 𝜓})) |
7 | 6 | eqrdv 2733 | 1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 [wsb 2062 ∈ wcel 2106 {cab 2712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: eqabcdv 2874 eqabi 2875 sbab 2887 rabeqcda 3445 iftrue 4537 iffalse 4540 dfopif 4875 iniseg 6118 setlikespec 6348 fncnvima2 7081 isoini 7358 dftpos3 8268 mapsnd 8925 hartogslem1 9580 r1val2 9875 cardval2 10029 dfac3 10159 wrdval 14552 wrdnval 14580 submgmacs 18743 submacs 18853 ablsimpgfind 20145 dfrhm2 20491 lsppr 21110 rspsn 21361 znunithash 21601 tgval3 22986 txrest 23655 xkoptsub 23678 cnextf 24090 cnblcld 24811 shft2rab 25557 sca2rab 25561 renegscl 28445 grpoinvf 30561 elpjrn 32219 ofrn2 32657 ellcsrspsn 35626 neibastop3 36345 ecres2 38261 lkrval2 39072 lshpset2N 39101 hdmapoc 41914 |
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