Theorem 2reu2rex 3377

Description: Double restricted existential uniqueness, analogous to 2eu2ex 2668. (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 3369	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
2		reurex 3369	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 𝜑)
3	2	reximi 3100	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	1, 3	syl 18	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   → wi 4  ∃wrex 3086  ∃!wreu 3363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-eu 2594  df-rex 3087  df-rmo 3365  df-reu 3366

This theorem is used by:  2reu1  3845  2sqreultblem  27687  2sqreunnltblem  27690