Theorem dftr2c 5206

Description: Variant of dftr2 5205 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2153. (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 5205	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
2		elequ1 2112	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
3	2	anbi1d 630	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
4		eleq1w 2819	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
5	3, 4	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)))
6		elequ2 2120	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
7		eleq1w 2819	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
8	6, 7	anbi12d 631	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴)))
9	8	imbi1d 341	. . 3 ⊢ (𝑦 = 𝑧 → (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ 𝐴)))
10	5, 9	alcomw 2046	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
11	1, 10	bitri 274	1 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1538   ∈ wcel 2105  Tr wtr 5203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3442  df-in 3903  df-ss 3913  df-uni 4850  df-tr 5204

This theorem is referenced by:  dftr5  5207