Theorem 2reu2rex 39832

Description: Double restricted existential uniqueness, analogous to 2eu2ex 2534. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Assertion

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

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reu2rex

Step	Hyp	Ref	Expression
1		reurex 3137	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
2		reurex 3137	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 𝜑)
3	2	reximi 2994	. 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 2897  ∃!wreu 2898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-eu 2462  df-mo 2463  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904

This theorem is referenced by:  2reu1  39835