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Mirrors > Home > ILE Home > Th. List > ralsn | GIF version |
Description: Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.) |
Ref | Expression |
---|---|
ralsn.1 | ⊢ 𝐴 ∈ V |
ralsn.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralsn | ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ralsn.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ralsng 3571 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 {csn 3532 |
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-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-sbc 2914 df-sn 3538 |
This theorem is referenced by: tfr0dm 6227 elixpsn 6637 finomni 7020 nninfsellemdc 13381 |
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