Theorem dford3lem1 42512

Description: Lemma for dford3 42514. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Assertion

Ref	Expression
dford3lem1	⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))

Distinct variable group:   𝑦,𝑏,𝑁

Proof of Theorem dford3lem1

Step	Hyp	Ref	Expression
1		treq 5268	. . . . 5 ⊢ (𝑦 = 𝑏 → (Tr 𝑦 ↔ Tr 𝑏))
2	1	cbvralvw 3225	. . . 4 ⊢ (∀𝑦 ∈ 𝑁 Tr 𝑦 ↔ ∀𝑏 ∈ 𝑁 Tr 𝑏)
3	2	biimpi 215	. . 3 ⊢ (∀𝑦 ∈ 𝑁 Tr 𝑦 → ∀𝑏 ∈ 𝑁 Tr 𝑏)
4	3	adantl 480	. 2 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 Tr 𝑏)
5		trss 5271	. . . . . 6 ⊢ (Tr 𝑁 → (𝑏 ∈ 𝑁 → 𝑏 ⊆ 𝑁))
6		ssralv 4041	. . . . . 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 405	. . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → (𝑏 ∈ 𝑁 → ∀𝑦 ∈ 𝑏 Tr 𝑦))
10	9	ralrimiv 3135	. 2 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 ∀𝑦 ∈ 𝑏 Tr 𝑦)
11		r19.26 3101	. 2 ⊢ (∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) ↔ (∀𝑏 ∈ 𝑁 Tr 𝑏 ∧ ∀𝑏 ∈ 𝑁 ∀𝑦 ∈ 𝑏 Tr 𝑦))
12	4, 10, 11	sylanbrc 581	1 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  ∀wral 3051   ⊆ wss 3939  Tr wtr 5260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-v 3465  df-ss 3956  df-uni 4904  df-tr 5261

This theorem is referenced by:  dford3lem2  42513  dford3  42514