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Mirrors > Home > ILE Home > Th. List > vtoclg | GIF version |
Description: Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) |
Ref | Expression |
---|---|
vtoclg.1 | ⊢ (x = A → (φ ↔ ψ)) |
vtoclg.2 | ⊢ φ |
Ref | Expression |
---|---|
vtoclg | ⊢ (A ∈ 𝑉 → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2175 | . 2 ⊢ ℲxA | |
2 | nfv 1418 | . 2 ⊢ Ⅎxψ | |
3 | vtoclg.1 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
4 | vtoclg.2 | . 2 ⊢ φ | |
5 | 1, 2, 3, 4 | vtoclgf 2606 | 1 ⊢ (A ∈ 𝑉 → ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 |
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-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 |
This theorem is referenced by: vtoclbg 2608 ceqex 2665 mo2icl 2714 nelrdva 2740 sbctt 2818 sbcnestgf 2891 csbing 3138 prnzg 3483 sneqrg 3524 unisng 3588 csbopabg 3826 trss 3854 inex1g 3884 ssexg 3887 pwexg 3924 prexgOLD 3937 prexg 3938 opth 3965 ordelord 4084 uniexg 4141 vtoclr 4331 resieq 4565 csbima12g 4629 dmsnsnsng 4741 iota5 4830 csbiotag 4838 funmo 4860 fconstg 5026 funfveu 5131 funbrfv 5155 fnbrfvb 5157 fvelimab 5172 ssimaexg 5178 fvelrn 5241 isoselem 5402 csbriotag 5423 csbov123g 5485 ovg 5581 tfrexlem 5889 rdg0g 5915 ensn1g 6213 fundmeng 6223 xpdom2g 6242 prcdnql 6467 prcunqu 6468 prdisj 6475 bdzfauscl 9345 bdinex1g 9356 bdssexg 9359 bj-prexg 9366 bj-uniexg 9373 |
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