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Mirrors > Home > MPE Home > Th. List > dftr2c | Structured version Visualization version GIF version |
Description: Variant of dftr2 5267 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2155. (Contributed by BTernaryTau, 28-Dec-2024.) |
Ref | Expression |
---|---|
dftr2c | ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr2 5267 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) | |
2 | elequ1 2113 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)) | |
3 | 2 | anbi1d 631 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
4 | eleq1w 2822 | . . . 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 2121 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧)) | |
7 | eleq1w 2822 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
8 | 6, 7 | anbi12d 632 | . . . 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 2042 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
11 | 1, 10 | bitri 275 | 1 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 ∈ wcel 2106 Tr wtr 5265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-uni 4913 df-tr 5266 |
This theorem is referenced by: dftr5 5269 |
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