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Mirrors > Home > MPE Home > Th. List > abbi2dv | Structured version Visualization version GIF version |
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
Ref | Expression |
---|---|
abbi2dv.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Ref | Expression |
---|---|
abbi2dv | ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi2dv.1 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) | |
2 | 1 | alrimiv 1842 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓)) |
3 | abeq2 2719 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓)) | |
4 | 2, 3 | sylibr 223 | 1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 {cab 2596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: abbi1dv 2730 sbab 2737 iftrue 4042 iffalse 4045 dfopif 4337 iniseg 5415 setlikespec 5618 fncnvima2 6247 isoini 6488 dftpos3 7257 hartogslem1 8330 r1val2 8583 cardval2 8700 dfac3 8827 wrdval 13163 wrdnval 13190 submacs 17188 dfrhm2 18540 lsppr 18914 rspsn 19075 znunithash 19732 tgval3 20578 txrest 21244 xkoptsub 21267 cnextf 21680 cnblcld 22388 shft2rab 23083 sca2rab 23087 grpoinvf 26770 elpjrn 28433 ofrn2 28822 neibastop3 31527 lkrval2 33395 lshpset2N 33424 hdmapoc 36241 mapsnd 38383 submgmacs 41594 |
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