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Mirrors > Home > ILE Home > Th. List > elab | GIF version |
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
elab.1 | ⊢ A ∈ V |
elab.2 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
elab | ⊢ (A ∈ {x ∣ φ} ↔ ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1418 | . 2 ⊢ Ⅎxψ | |
2 | elab.1 | . 2 ⊢ A ∈ V | |
3 | elab.2 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
4 | 1, 2, 3 | elabf 2680 | 1 ⊢ (A ∈ {x ∣ φ} ↔ ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {cab 2023 Vcvv 2551 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 |
This theorem is referenced by: ralab 2695 rexab 2697 intab 3635 dfiin2g 3681 dfiunv2 3684 uniuni 4149 peano5 4264 finds 4266 finds2 4267 funcnvuni 4911 fun11iun 5090 elabrex 5340 abrexco 5341 indpi 6326 nqprm 6525 nqprrnd 6526 nqprdisj 6527 nqprloc 6528 nqprl 6533 cauappcvgprlem2 6632 caucvgprlem2 6651 1nn 7706 peano2nn 7707 dfuzi 8124 bj-ssom 9395 |
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