Theorem reuanid 3256

Description: Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.)

Assertion

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

Proof of Theorem reuanid

Step	Hyp	Ref	Expression
1		anabs5 659	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	eubii 2585	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-reu 3070	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		df-reu 3070	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	2, 3, 4	3bitr4i 302	1 ⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∃!weu 2568  ∃!wreu 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-eu 2569  df-reu 3070

This theorem is referenced by: (None)