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Mirrors > Home > ILE Home > Th. List > rspc | GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) |
Ref | Expression |
---|---|
rspc.1 | ⊢ Ⅎ𝑥𝜓 |
rspc.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspc | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2375 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑)) | |
2 | nfcv 2235 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfv 1473 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | |
4 | rspc.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfim 1516 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓) |
6 | eleq1 2157 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
7 | rspc.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
8 | 6, 7 | imbi12d 233 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))) |
9 | 2, 5, 8 | spcgf 2715 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜑) → (𝐴 ∈ 𝐵 → 𝜓))) |
10 | 9 | pm2.43a 51 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜑) → 𝜓)) |
11 | 1, 10 | syl5bi 151 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1294 = wceq 1296 Ⅎwnf 1401 ∈ wcel 1445 ∀wral 2370 |
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-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-v 2635 |
This theorem is referenced by: rspcv 2732 rspc2 2746 pofun 4163 omsinds 4463 fmptcof 5504 fliftfuns 5615 qliftfuns 6416 xpf1o 6640 finexdc 6698 ssfirab 6723 iunfidisj 6735 lble 8505 exfzdc 9800 uzsinds 9997 sumeq2 10902 sumfct 10917 sumrbdclem 10919 summodclem3 10923 summodclem2a 10924 zsumdc 10927 fsumgcl 10929 fsum3 10930 fsumf1o 10933 isumss 10934 isumss2 10936 fsum3cvg2 10937 fsumadd 10949 isummulc2 10969 fsum2dlemstep 10977 fisumcom2 10981 fsumshftm 10988 fisum0diag2 10990 fsummulc2 10991 fsum00 11005 fsumabs 11008 fsumrelem 11014 fsumiun 11020 isumshft 11033 mertenslem2 11079 zsupcllemstep 11368 infssuzex 11372 bezoutlemmain 11414 iuncld 11967 bj-nntrans 12554 |
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