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| Mirrors > Home > MPE Home > Th. List > Mathboxes > reutruALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of reutru 48724. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| reutruALT | ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rextru 3077 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤) | |
| 2 | rmotru 48723 | . . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤) | |
| 3 | 1, 2 | anbi12i 628 | . 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐴 ⊤ ∧ ∃*𝑥 ∈ 𝐴 ⊤)) |
| 4 | df-eu 2569 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴)) | |
| 5 | reu5 3382 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 ⊤ ↔ (∃𝑥 ∈ 𝐴 ⊤ ∧ ∃*𝑥 ∈ 𝐴 ⊤)) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ⊤wtru 1541 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2538 ∃!weu 2568 ∃wrex 3070 ∃!wreu 3378 ∃*wrmo 3379 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-mo 2540 df-eu 2569 df-rex 3071 df-rmo 3380 df-reu 3381 |
| This theorem is referenced by: (None) |
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