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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 2986 | . . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ 𝑇 𝜑 ↔ ∀𝑧 ∈ 𝑇 𝜒)) |
3 | rspc3v.2 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) | |
4 | 3 | ralbidv 2986 | . . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝑇 𝜒 ↔ ∀𝑧 ∈ 𝑇 𝜃)) |
5 | 2, 4 | rspc2v 3322 | . . 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 3305 | . . 3 ⊢ (𝐶 ∈ 𝑇 → (∀𝑧 ∈ 𝑇 𝜃 → 𝜓)) |
8 | 5, 7 | sylan9 689 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓)) |
9 | 8 | 3impa 1259 | 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 |
This theorem is referenced by: swopolem 5044 isopolem 6595 caovassg 6832 caovcang 6835 caovordig 6839 caovordg 6841 caovdig 6848 caovdirg 6851 caofass 6931 caoftrn 6932 prslem 16931 posi 16950 latdisdlem 17189 dlatmjdi 17194 sgrpass 17290 gaass 17730 islmodd 18869 rmodislmodlem 18930 rmodislmod 18931 lsscl 18943 assalem 19316 psmettri2 22114 xmettri2 22145 axtgcgrid 25362 axtg5seg 25364 axtgpasch 25366 axtgupdim2 25370 axtgeucl 25371 tgdim01 25402 f1otrgitv 25750 grpoass 27357 vcdi 27420 vcdir 27421 vcass 27422 lnolin 27609 lnopl 28773 lnfnl 28790 omndadd 29706 axtgupdim2OLD 30746 rngodi 33703 rngodir 33704 rngoass 33705 lfli 34348 cvlexch1 34615 rngdir 41882 |
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