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Mirrors > Home > MPE Home > Th. List > rspc3v | Structured version Visualization version GIF version |
Description: 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.) |
Ref | Expression |
---|---|
rspc3v.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
rspc3v.2 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) |
rspc3v.3 | ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspc3v | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspc3v.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
2 | 1 | ralbidv 2969 | . . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ 𝑇 𝜑 ↔ ∀𝑧 ∈ 𝑇 𝜒)) |
3 | rspc3v.2 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) | |
4 | 3 | ralbidv 2969 | . . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝑇 𝜒 ↔ ∀𝑧 ∈ 𝑇 𝜃)) |
5 | 2, 4 | rspc2v 3293 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → ∀𝑧 ∈ 𝑇 𝜃)) |
6 | rspc3v.3 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓)) | |
7 | 6 | rspcv 3278 | . . 3 ⊢ (𝐶 ∈ 𝑇 → (∀𝑧 ∈ 𝑇 𝜃 → 𝜓)) |
8 | 5, 7 | sylan9 687 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓)) |
9 | 8 | 3impa 1251 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 |
This theorem is referenced by: swopolem 4968 isopolem 6495 caovassg 6730 caovcang 6733 caovordig 6737 caovordg 6739 caovdig 6746 caovdirg 6749 caofass 6829 caoftrn 6830 prslem 16754 posi 16773 latdisdlem 17012 dlatmjdi 17017 sgrpass 17113 gaass 17553 islmodd 18692 lsscl 18764 assalem 19137 psmettri2 21924 xmettri2 21955 axtgcgrid 25162 axtg5seg 25164 axtgpasch 25166 axtgupdim2 25170 axtgeucl 25171 tgdim01 25202 f1otrgitv 25550 grpoass 26741 vcdi 26804 vcdir 26805 vcass 26806 lnolin 26993 lnopl 28157 lnfnl 28174 omndadd 29037 axtgupdim2OLD 29999 rngodi 32873 rngodir 32874 rngoass 32875 lfli 33366 cvlexch1 33633 rngdir 41672 |
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