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Theorem reutruALT 46152
Description: Alternate proof for reutru 46151. (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 46149 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑥𝐴 ⊤)
2 rmotru 46150 . . 3 (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥𝐴 ⊤)
31, 2anbi12i 627 . 2 ((∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴) ↔ (∃𝑥𝐴 ⊤ ∧ ∃*𝑥𝐴 ⊤))
4 df-eu 2569 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴))
5 reu5 3361 . 2 (∃!𝑥𝐴 ⊤ ↔ (∃𝑥𝐴 ⊤ ∧ ∃*𝑥𝐴 ⊤))
63, 4, 53bitr4i 303 1 (∃!𝑥 𝑥𝐴 ↔ ∃!𝑥𝐴 ⊤)
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)
  Copyright terms: Public domain W3C validator