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Mirrors > Home > MPE Home > Th. List > Mathboxes > dford3lem1 | Structured version Visualization version GIF version |
Description: Lemma for dford3 39623. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
Ref | Expression |
---|---|
dford3lem1 | ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | treq 5177 | . . . . 5 ⊢ (𝑦 = 𝑏 → (Tr 𝑦 ↔ Tr 𝑏)) | |
2 | 1 | cbvralvw 3449 | . . . 4 ⊢ (∀𝑦 ∈ 𝑁 Tr 𝑦 ↔ ∀𝑏 ∈ 𝑁 Tr 𝑏) |
3 | 2 | biimpi 218 | . . 3 ⊢ (∀𝑦 ∈ 𝑁 Tr 𝑦 → ∀𝑏 ∈ 𝑁 Tr 𝑏) |
4 | 3 | adantl 484 | . 2 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 Tr 𝑏) |
5 | trss 5180 | . . . . . 6 ⊢ (Tr 𝑁 → (𝑏 ∈ 𝑁 → 𝑏 ⊆ 𝑁)) | |
6 | ssralv 4032 | . . . . . 6 ⊢ (𝑏 ⊆ 𝑁 → (∀𝑦 ∈ 𝑁 Tr 𝑦 → ∀𝑦 ∈ 𝑏 Tr 𝑦)) | |
7 | 5, 6 | syl6 35 | . . . . 5 ⊢ (Tr 𝑁 → (𝑏 ∈ 𝑁 → (∀𝑦 ∈ 𝑁 Tr 𝑦 → ∀𝑦 ∈ 𝑏 Tr 𝑦))) |
8 | 7 | com23 86 | . . . 4 ⊢ (Tr 𝑁 → (∀𝑦 ∈ 𝑁 Tr 𝑦 → (𝑏 ∈ 𝑁 → ∀𝑦 ∈ 𝑏 Tr 𝑦))) |
9 | 8 | imp 409 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → (𝑏 ∈ 𝑁 → ∀𝑦 ∈ 𝑏 Tr 𝑦)) |
10 | 9 | ralrimiv 3181 | . 2 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 ∀𝑦 ∈ 𝑏 Tr 𝑦) |
11 | r19.26 3170 | . 2 ⊢ (∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) ↔ (∀𝑏 ∈ 𝑁 Tr 𝑏 ∧ ∀𝑏 ∈ 𝑁 ∀𝑦 ∈ 𝑏 Tr 𝑦)) | |
12 | 4, 10, 11 | sylanbrc 585 | 1 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 Tr wtr 5171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-in 3942 df-ss 3951 df-uni 4838 df-tr 5172 |
This theorem is referenced by: dford3lem2 39622 dford3 39623 |
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