Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > rspc3v | 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 2457 | . . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ 𝑇 𝜑 ↔ ∀𝑧 ∈ 𝑇 𝜒)) |
3 | rspc3v.2 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) | |
4 | 3 | ralbidv 2457 | . . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝑇 𝜒 ↔ ∀𝑧 ∈ 𝑇 𝜃)) |
5 | 2, 4 | rspc2v 2829 | . . 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 2812 | . . 3 ⊢ (𝐶 ∈ 𝑇 → (∀𝑧 ∈ 𝑇 𝜃 → 𝜓)) |
8 | 5, 7 | sylan9 407 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓)) |
9 | 8 | 3impa 1177 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 ∀wral 2435 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 |
This theorem is referenced by: swopolem 4265 isopolem 5772 isosolem 5774 caovassg 5979 caovcang 5982 caovordig 5986 caovordg 5988 caovdig 5995 caovdirg 5998 caoftrn 6057 psmettri2 12739 xmettri2 12772 |
Copyright terms: Public domain | W3C validator |