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 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)

Proof of Theorem reutruALT

Step	Hyp	Ref	Expression
1		rextru 45728	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
2		rmotru 45729	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤)
3	1, 2	anbi12i 630	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐴 ⊤ ∧ ∃*𝑥 ∈ 𝐴 ⊤))
4		df-eu 2571	. 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴))
5		reu5 3329	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ⊤ ↔ (∃𝑥 ∈ 𝐴 ⊤ ∧ ∃*𝑥 ∈ 𝐴 ⊤))
6	3, 4, 5	3bitr4i 306	1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)

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)