Theorem reutruALT 48538

Description: Alternate proof for reutru 48537. (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 3083	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
2		rmotru 48536	. . 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 2572	. 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴))
5		reu5 3390	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ⊤ ↔ (∃𝑥 ∈ 𝐴 ⊤ ∧ ∃*𝑥 ∈ 𝐴 ⊤))
6	3, 4, 5	3bitr4i 303	1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)

Syntax hints:   ↔ wb 206   ∧ wa 395  ⊤wtru 1538  ∃wex 1777   ∈ wcel 2108  ∃*wmo 2541  ∃!weu 2571  ∃wrex 3076  ∃!wreu 3386  ∃*wrmo 3387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-mo 2543  df-eu 2572  df-rex 3077  df-rmo 3388  df-reu 3389

This theorem is referenced by: (None)