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Theorem dftr2c 5262
Description: Variant of dftr2 5261 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2157. (Contributed by BTernaryTau, 28-Dec-2024.)
Assertion
Ref Expression
dftr2c (Tr 𝐴 ↔ ∀𝑦𝑥((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem dftr2c
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dftr2 5261 . 2 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
2 elequ1 2115 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑦𝑧𝑦))
32anbi1d 631 . . . 4 (𝑥 = 𝑧 → ((𝑥𝑦𝑦𝐴) ↔ (𝑧𝑦𝑦𝐴)))
4 eleq1w 2824 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
53, 4imbi12d 344 . . 3 (𝑥 = 𝑧 → (((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
6 elequ2 2123 . . . . 5 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
7 eleq1w 2824 . . . . 5 (𝑦 = 𝑧 → (𝑦𝐴𝑧𝐴))
86, 7anbi12d 632 . . . 4 (𝑦 = 𝑧 → ((𝑥𝑦𝑦𝐴) ↔ (𝑥𝑧𝑧𝐴)))
98imbi1d 341 . . 3 (𝑦 = 𝑧 → (((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ((𝑥𝑧𝑧𝐴) → 𝑥𝐴)))
105, 9alcomw 2044 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ∀𝑦𝑥((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
111, 10bitri 275 1 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  Tr wtr 5259
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260
This theorem is referenced by:  dftr5  5263
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