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Mirrors > Home > MPE Home > Th. List > Mathboxes > reutruALT | Structured version Visualization version GIF version |
Description: Alternate proof for reutru 46151. (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 46149 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤) | |
2 | rmotru 46150 | . . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤) | |
3 | 1, 2 | anbi12i 627 | . 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 3361 | . 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 205 ∧ wa 396 ⊤wtru 1540 ∃wex 1782 ∈ wcel 2106 ∃*wmo 2538 ∃!weu 2568 ∃wrex 3065 ∃!wreu 3066 ∃*wrmo 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-mo 2540 df-eu 2569 df-rex 3070 df-rmo 3071 df-reu 3072 |
This theorem is referenced by: (None) |
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