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Mirrors > Home > ILE Home > Th. List > vtocl | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
vtocl.1 | ⊢ 𝐴 ∈ V |
vtocl.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl.3 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | vtocl.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | vtocl.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | vtocl.3 | . 2 ⊢ 𝜑 | |
5 | 1, 2, 3, 4 | vtoclf 2607 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: vtoclb 2611 zfauscl 3877 bnd2 3926 pwex 3932 uniex 4174 ordtriexmid 4247 onsucsssucexmid 4252 regexmid 4260 ordsoexmid 4286 onintexmid 4297 reg3exmid 4304 nnregexmid 4342 acexmidlemv 5510 caovcan 5665 findcard2 6346 findcard2s 6347 bj-uniex 10037 bj-omtrans 10081 |
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