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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 1476 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | elab.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | elab.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | elabf 2781 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1299 ∈ wcel 1448 {cab 2086 Vcvv 2641 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 |
This theorem is referenced by: ralab 2797 rexab 2799 intab 3747 dfiin2g 3793 dfiunv2 3796 uniuni 4310 dcextest 4433 peano5 4450 finds 4452 finds2 4453 funcnvuni 5128 fun11iun 5322 elabrex 5591 abrexco 5592 mapval2 6502 ssenen 6674 snexxph 6766 sbthlem2 6774 indpi 7051 nqprm 7251 nqprrnd 7252 nqprdisj 7253 nqprloc 7254 nqprl 7260 nqpru 7261 cauappcvgprlem2 7369 caucvgprlem2 7389 peano1nnnn 7539 peano2nnnn 7540 1nn 8589 peano2nn 8590 dfuzi 9013 hashfacen 10420 shftfvalg 10431 ovshftex 10432 shftfval 10434 txdis1cn 12228 bj-ssom 12719 |
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