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Mirrors > Home > ILE Home > Th. List > rgen2a | GIF version |
Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2004. Usage of rgen2 2550 instead is highly encouraged. (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rgen2a.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑) |
Ref | Expression |
---|---|
rgen2a | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 | |
2 | eleq1 2227 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | dvelimor 2005 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴) |
4 | eleq1 2227 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
5 | rgen2a.1 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑) | |
6 | 5 | ex 114 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝜑)) |
7 | 4, 6 | syl6bi 162 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝜑))) |
8 | 7 | pm2.43d 50 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜑)) |
9 | 8 | alimi 1442 | . . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) |
10 | 9 | a1d 22 | . . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))) |
11 | nfr 1505 | . . . . . 6 ⊢ (Ⅎ𝑦 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)) | |
12 | 6 | alimi 1442 | . . . . . 6 ⊢ (∀𝑦 𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) |
13 | 11, 12 | syl6 33 | . . . . 5 ⊢ (Ⅎ𝑦 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))) |
14 | 10, 13 | jaoi 706 | . . . 4 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))) |
15 | 3, 14 | ax-mp 5 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) |
16 | df-ral 2447 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) | |
17 | 15, 16 | sylibr 133 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) |
18 | 17 | rgen 2517 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 698 ∀wal 1340 = wceq 1342 Ⅎwnf 1447 ∈ wcel 2135 ∀wral 2442 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-cleq 2157 df-clel 2160 df-ral 2447 |
This theorem is referenced by: ordsucunielexmid 4502 onintexmid 4544 isoid 5772 issmo 6247 oawordriexmid 6429 ecopover 6590 ecopoverg 6593 1domsn 6776 unfiexmid 6874 axaddf 7800 axmulf 7801 subf 8091 negiso 8841 cnref1o 9579 xaddf 9771 ioof 9898 fzof 10069 xrnegiso 11189 reeff1 11627 gcdf 11890 eucalgf 11966 qredeu 12008 qnnen 12307 strsetsid 12370 hmeofn 12849 ismeti 12893 qtopbasss 13068 tgqioo 13094 peano4nninf 13727 |
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