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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 3178 | . . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ 𝑇 𝜑 ↔ ∀𝑧 ∈ 𝑇 𝜒)) |
3 | rspc3v.2 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) | |
4 | 3 | ralbidv 3178 | . . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝑇 𝜒 ↔ ∀𝑧 ∈ 𝑇 𝜃)) |
5 | 2, 4 | rspc2v 3623 | . . 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 3609 | . . 3 ⊢ (𝐶 ∈ 𝑇 → (∀𝑧 ∈ 𝑇 𝜃 → 𝜓)) |
8 | 5, 7 | sylan9 509 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓)) |
9 | 8 | 3impa 1111 | 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 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 |
This theorem is referenced by: rspc3dv 3630 rspc4v 3631 pocl 5596 swopolem 5599 isopolem 7342 caovassg 7605 caovcang 7608 caovordig 7612 caovordg 7614 caovdig 7621 caovdirg 7624 caofass 7707 caoftrn 7708 frpoins3xp3g 8127 prslem 18251 posi 18270 latdisdlem 18449 dlatmjdi 18476 sgrpass 18616 gaass 19161 o2timesd 20033 rglcom4d 20034 islmodd 20477 rmodislmodlem 20539 rmodislmod 20540 rmodislmodOLD 20541 lsscl 20553 assalem 21412 psmettri2 23815 xmettri2 23846 addsproplem1 27453 addsprop 27460 axtgcgrid 27714 axtg5seg 27716 axtgpasch 27718 axtgupdim2 27722 axtgeucl 27723 tgdim01 27758 f1otrgitv 28121 grpoass 29756 vcdi 29818 vcdir 29819 vcass 29820 lnolin 30007 lnopl 31167 lnfnl 31184 omndadd 32224 axtgupdim2ALTV 33680 rngodi 36772 rngodir 36773 rngoass 36774 lfli 37931 cvlexch1 38198 rngdi 46659 rngdir 46660 isthincd2lem2 47656 |
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