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| Mirrors > Home > MPE Home > Th. List > dftr2c | Structured version Visualization version GIF version | ||
| Description: Variant of dftr2 5261 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2157. (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 5261 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) | |
| 2 | elequ1 2115 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)) | |
| 3 | 2 | anbi1d 631 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
| 4 | eleq1w 2824 | . . . 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 2123 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧)) | |
| 7 | eleq1w 2824 | . . . . 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 2044 | . 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 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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