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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 1473 | . 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 2686 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1296 ∈ wcel 1445 Vcvv 2633 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-v 2635 |
This theorem is referenced by: vtoclb 2690 zfauscl 3980 bnd2 4029 uniex 4288 ordtriexmid 4366 onsucsssucexmid 4371 regexmid 4379 ordsoexmid 4406 onintexmid 4416 reg3exmid 4423 nnregexmid 4462 acexmidlemv 5688 caovcan 5847 findcard2 6685 findcard2s 6686 inffiexmid 6702 sup3exmid 8515 bj-uniex 12532 bj-omtrans 12575 |
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