Theorem dftr5OLD 5203

Description: Obsolete version of dftr5 5202 as of 28-Dec-2024. (Contributed by NM, 20-Mar-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dftr5OLD	⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem dftr5OLD

Step	Hyp	Ref	Expression
1		dftr2 5200	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
2		alcom 2154	. . 3 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
3		impexp 452	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)))
4	3	albii 1819	. . . . . . 7 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)))
5		df-ral 3063	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)))
6	4, 5	bitr4i 278	. . . . . 6 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ∀𝑦 ∈ 𝑥 (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
7		r19.21v 3173	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴))
8	6, 7	bitri 275	. . . . 5 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴))
9	8	albii 1819	. . . 4 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴))
10		df-ral 3063	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴))
11	9, 10	bitr4i 278	. . 3 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
12	2, 11	bitri 275	. 2 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
13	1, 12	bitri 275	1 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1537   ∈ wcel 2104  ∀wral 3062  Tr wtr 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-v 3439  df-in 3899  df-ss 3909  df-uni 4845  df-tr 5199

This theorem is referenced by: (None)