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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 1539 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | elab.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | elab.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | elabf 2895 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 {cab 2175 Vcvv 2752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 |
This theorem is referenced by: ralab 2912 rexab 2914 intab 3891 dfiin2g 3937 dfiunv2 3940 uniuni 4472 dcextest 4601 peano5 4618 finds 4620 finds2 4621 funcnvuni 5307 fun11iun 5504 elabrex 5782 abrexco 5784 mapval2 6708 ssenen 6883 snexxph 6983 sbthlem2 6991 indpi 7376 nqprm 7576 nqprrnd 7577 nqprdisj 7578 nqprloc 7579 nqprl 7585 nqpru 7586 cauappcvgprlem2 7694 caucvgprlem2 7714 peano1nnnn 7886 peano2nnnn 7887 1nn 8965 peano2nn 8966 dfuzi 9398 hashfacen 10857 shftfvalg 10868 ovshftex 10869 shftfval 10871 4sqlemafi 12438 lss1d 13724 txdis1cn 14263 bj-ssom 15174 |
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