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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 1461 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | elab.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | elab.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | elabf 2737 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 {cab 2067 Vcvv 2601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 |
This theorem is referenced by: ralab 2752 rexab 2754 intab 3665 dfiin2g 3711 dfiunv2 3714 uniuni 4201 peano5 4339 finds 4341 finds2 4342 funcnvuni 4988 fun11iun 5167 elabrex 5418 abrexco 5419 indpi 6532 nqprm 6732 nqprrnd 6733 nqprdisj 6734 nqprloc 6735 nqprl 6741 nqpru 6742 cauappcvgprlem2 6850 caucvgprlem2 6870 peano1nnnn 7020 peano2nnnn 7021 1nn 8050 peano2nn 8051 dfuzi 8457 shftfvalg 9706 ovshftex 9707 shftfval 9709 bj-ssom 10731 |
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