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Mirrors > Home > ILE Home > Th. List > rgen2a | GIF version |
Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct (and illustrates the use of dvelimor 1993). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.) |
Ref | Expression |
---|---|
rgen2a.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑) |
Ref | Expression |
---|---|
rgen2a | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 | |
2 | eleq1 2202 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | dvelimor 1993 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴) |
4 | eleq1 2202 | . . . . . . . . 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 1431 | . . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) |
10 | 9 | a1d 22 | . . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))) |
11 | nfr 1498 | . . . . . 6 ⊢ (Ⅎ𝑦 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)) | |
12 | 6 | alimi 1431 | . . . . . 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 705 | . . . 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 2421 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) | |
17 | 15, 16 | sylibr 133 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) |
18 | 17 | rgen 2485 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 697 ∀wal 1329 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 ∀wral 2416 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-cleq 2132 df-clel 2135 df-ral 2421 |
This theorem is referenced by: ordsucunielexmid 4446 onintexmid 4487 isoid 5711 issmo 6185 oawordriexmid 6366 ecopover 6527 ecopoverg 6530 1domsn 6713 unfiexmid 6806 axaddf 7683 axmulf 7684 subf 7971 negiso 8720 cnref1o 9447 xaddf 9634 ioof 9761 fzof 9928 xrnegiso 11038 reeff1 11414 gcdf 11668 eucalgf 11743 qredeu 11785 qnnen 11951 strsetsid 12002 hmeofn 12481 ismeti 12525 qtopbasss 12700 tgqioo 12726 peano4nninf 13210 |
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