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Mirrors > Home > ILE Home > Th. List > vtoclga | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.) |
Ref | Expression |
---|---|
vtoclga.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtoclga.2 | ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
Ref | Expression |
---|---|
vtoclga | ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | vtoclga.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | vtoclga.2 | . 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
5 | 1, 2, 3, 4 | vtoclgaf 2618 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: vtoclri 2628 ssuni 3602 ordtriexmid 4247 onsucsssucexmid 4252 tfis3 4309 fvmpt3 5251 fvmptssdm 5255 fnressn 5349 fressnfv 5350 caovord 5672 caovimo 5694 tfrlem1 5923 freccl 5993 nnacl 6059 nnmcl 6060 nnacom 6063 nnaass 6064 nndi 6065 nnmass 6066 nnmsucr 6067 nnmcom 6068 nnsucsssuc 6071 nntri3or 6072 nnaordi 6081 nnaword 6084 nnmordi 6089 nnaordex 6100 findcard 6345 findcard2 6346 findcard2s 6347 indpi 6440 prarloclem3 6595 uzind4s2 8534 cnref1o 8582 frec2uzrdg 9195 expcl2lemap 9267 climub 9864 climserile 9865 ialginv 9886 ialgcvg 9887 ialgcvga 9890 ialgfx 9891 |
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