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Theorem reutruALT 45731
Description: Alternate proof for reutru 45730. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
reutruALT (∃!𝑥 𝑥𝐴 ↔ ∃!𝑥𝐴 ⊤)

Proof of Theorem reutruALT
StepHypRef Expression
1 rextru 45728 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑥𝐴 ⊤)
2 rmotru 45729 . . 3 (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥𝐴 ⊤)
31, 2anbi12i 630 . 2 ((∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴) ↔ (∃𝑥𝐴 ⊤ ∧ ∃*𝑥𝐴 ⊤))
4 df-eu 2571 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴))
5 reu5 3329 . 2 (∃!𝑥𝐴 ⊤ ↔ (∃𝑥𝐴 ⊤ ∧ ∃*𝑥𝐴 ⊤))
63, 4, 53bitr4i 306 1 (∃!𝑥 𝑥𝐴 ↔ ∃!𝑥𝐴 ⊤)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wtru 1543  wex 1786  wcel 2114  ∃*wmo 2539  ∃!weu 2570  wrex 3055  ∃!wreu 3056  ∃*wrmo 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-mo 2541  df-eu 2571  df-rex 3060  df-reu 3061  df-rmo 3062
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator