Theorem dford3lem1 38378

Description: Lemma for dford3 38380. (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 4951	. . . . 5 ⊢ (𝑦 = 𝑏 → (Tr 𝑦 ↔ Tr 𝑏))
2	1	cbvralv 3354	. . . 4 ⊢ (∀𝑦 ∈ 𝑁 Tr 𝑦 ↔ ∀𝑏 ∈ 𝑁 Tr 𝑏)
3	2	biimpi 208	. . 3 ⊢ (∀𝑦 ∈ 𝑁 Tr 𝑦 → ∀𝑏 ∈ 𝑁 Tr 𝑏)
4	3	adantl 474	. 2 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 Tr 𝑏)
5		trss 4954	. . . . . 6 ⊢ (Tr 𝑁 → (𝑏 ∈ 𝑁 → 𝑏 ⊆ 𝑁))
6		ssralv 3862	. . . . . 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 396	. . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → (𝑏 ∈ 𝑁 → ∀𝑦 ∈ 𝑏 Tr 𝑦))
10	9	ralrimiv 3146	. 2 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 ∀𝑦 ∈ 𝑏 Tr 𝑦)
11		r19.26 3245	. 2 ⊢ (∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) ↔ (∀𝑏 ∈ 𝑁 Tr 𝑏 ∧ ∀𝑏 ∈ 𝑁 ∀𝑦 ∈ 𝑏 Tr 𝑦))
12	4, 10, 11	sylanbrc 579	1 ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))

Syntax hints:   → wi 4   ∧ wa 385   ∈ wcel 2157  ∀wral 3089   ⊆ wss 3769  Tr wtr 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-in 3776  df-ss 3783  df-uni 4629  df-tr 4946

This theorem is referenced by:  dford3lem2  38379  dford3  38380