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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 2258 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1493 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | vtoclga.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | vtoclga.2 | . 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
5 | 1, 2, 3, 4 | vtoclgaf 2725 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 |
This theorem is referenced by: vtoclri 2735 ssuni 3728 ordtriexmid 4407 onsucsssucexmid 4412 tfis3 4470 fvmpt3 5468 fvmptssdm 5473 fnressn 5574 fressnfv 5575 caovord 5910 caovimo 5932 tfrlem1 6173 nnacl 6344 nnmcl 6345 nnacom 6348 nnaass 6349 nndi 6350 nnmass 6351 nnmsucr 6352 nnmcom 6353 nnsucsssuc 6356 nntri3or 6357 nnaordi 6372 nnaword 6375 nnmordi 6380 nnaordex 6391 ixpfn 6566 findcard 6750 findcard2 6751 findcard2s 6752 exmidomni 6982 indpi 7118 prarloclem3 7273 uzind4s2 9354 cnref1o 9408 frec2uzrdg 10150 expcl2lemap 10273 seq3coll 10553 climub 11081 climserle 11082 fsum3cvg 11114 summodclem2a 11118 alginv 11655 algcvg 11656 algcvga 11659 algfx 11660 prmind2 11728 |
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