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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 1509 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | elab.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | elab.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | elabf 2831 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 {cab 2126 Vcvv 2689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 |
This theorem is referenced by: ralab 2848 rexab 2850 intab 3808 dfiin2g 3854 dfiunv2 3857 uniuni 4380 dcextest 4503 peano5 4520 finds 4522 finds2 4523 funcnvuni 5200 fun11iun 5396 elabrex 5667 abrexco 5668 mapval2 6580 ssenen 6753 snexxph 6846 sbthlem2 6854 indpi 7174 nqprm 7374 nqprrnd 7375 nqprdisj 7376 nqprloc 7377 nqprl 7383 nqpru 7384 cauappcvgprlem2 7492 caucvgprlem2 7512 peano1nnnn 7684 peano2nnnn 7685 1nn 8755 peano2nn 8756 dfuzi 9185 hashfacen 10611 shftfvalg 10622 ovshftex 10623 shftfval 10625 txdis1cn 12486 bj-ssom 13305 |
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