Theorem 2reu2rex 3353

Description: Double restricted existential uniqueness, analogous to 2eu2ex 2645. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Assertion

Ref	Expression
2reu2rex	⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Proof of Theorem 2reu2rex

Step	Hyp	Ref	Expression
1		reurex 3352	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
2		reurex 3352	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 𝜑)
3	2	reximi 3174	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	1, 3	syl 17	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4  ∃wrex 3064  ∃!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-eu 2569  df-rex 3069  df-reu 3070  df-rmo 3071

This theorem is referenced by:  2reu1  3826  2sqreultblem  26501  2sqreunnltblem  26504