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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 2258 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1493 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | vtoclg.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | vtoclg.2 | . 2 ⊢ 𝜑 | |
5 | 1, 2, 3, 4 | vtoclgf 2718 | 1 ⊢ (𝐴 ∈ 𝑉 → 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: vtoclbg 2721 ceqex 2786 mo2icl 2836 nelrdva 2864 sbctt 2947 sbcnestgf 3021 csbing 3253 ifmdc 3479 prnzg 3617 sneqrg 3659 unisng 3723 csbopabg 3976 trss 4005 inex1g 4034 ssexg 4037 pwexg 4074 prexg 4103 opth 4129 ordelord 4273 uniexg 4331 vtoclr 4557 resieq 4799 csbima12g 4870 dmsnsnsng 4986 iota5 5078 csbiotag 5086 funmo 5108 fconstg 5289 funfveu 5402 funbrfv 5428 fnbrfvb 5430 fvelimab 5445 ssimaexg 5451 fvelrn 5519 isoselem 5689 csbriotag 5710 csbov123g 5777 ovg 5877 tfrexlem 6199 rdg0g 6253 ensn1g 6659 fundmeng 6669 xpdom2g 6694 phplem3g 6718 prcdnql 7260 prcunqu 7261 prdisj 7268 shftvalg 10576 shftval4g 10577 climshft 11041 telfsumo 11203 fsumparts 11207 lcmgcdlem 11685 fiinopn 12098 bdzfauscl 13015 bdinex1g 13026 bdssexg 13029 bj-prexg 13036 bj-uniexg 13043 |
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