Theorem dftr2c 5215

Description: Variant of dftr2 5214 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2194. (Contributed by BTernaryTau, 28-Dec-2024.)

Assertion

Ref	Expression
dftr2c	⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem dftr2c

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr2 5214	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
2		elequ1 2152	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
3	2	anbi1d 643	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
4		eleq1w 2844	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
5	3, 4	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)))
6		elequ2 2160	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
7		eleq1w 2844	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
8	6, 7	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴)))
9	8	imbi1d 344	. . 3 ⊢ (𝑦 = 𝑧 → (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ 𝐴)))
10	5, 9	alcomw 2078	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
11	1, 10	bitri 278	1 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  Tr wtr 5212

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-ss 3916  df-uni 4868  df-tr 5213

This theorem is used by:  dftr5  5216