Theorem reuanidOLD 3392

Description: Obsolete version of reuanid 3390 as of 12-Jan-2025. (Contributed by Peter Mazsa, 12-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
reuanidOLD	⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑)

Proof of Theorem reuanidOLD

Step	Hyp	Ref	Expression
1		anabs5 663	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	eubii 2584	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-reu 3380	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		df-reu 3380	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	2, 3, 4	3bitr4i 303	1 ⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2107  ∃!weu 2567  ∃!wreu 3377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-eu 2568  df-reu 3380

This theorem is referenced by: (None)