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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 1969). (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 1491 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 | |
2 | eleq1 2177 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | dvelimor 1969 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴) |
4 | eleq1 2177 | . . . . . . . . 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 1414 | . . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) |
10 | 9 | a1d 22 | . . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))) |
11 | nfr 1481 | . . . . . 6 ⊢ (Ⅎ𝑦 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)) | |
12 | 6 | alimi 1414 | . . . . . 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 688 | . . . 4 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))) |
15 | 3, 14 | ax-mp 7 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) |
16 | df-ral 2395 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) | |
17 | 15, 16 | sylibr 133 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) |
18 | 17 | rgen 2459 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 680 ∀wal 1312 = wceq 1314 Ⅎwnf 1419 ∈ wcel 1463 ∀wral 2390 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-nf 1420 df-sb 1719 df-cleq 2108 df-clel 2111 df-ral 2395 |
This theorem is referenced by: ordsucunielexmid 4406 onintexmid 4447 isoid 5665 issmo 6139 oawordriexmid 6320 ecopover 6481 ecopoverg 6484 1domsn 6666 unfiexmid 6759 axaddf 7603 axmulf 7604 subf 7887 negiso 8623 cnref1o 9342 xaddf 9520 ioof 9647 fzof 9814 xrnegiso 10923 reeff1 11258 gcdf 11509 eucalgf 11582 qredeu 11624 qnnen 11789 strsetsid 11835 ismeti 12335 qtopbasss 12510 tgqioo 12533 peano4nninf 12892 |
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