Theorem 2rexreu 42011

Description: Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2730. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Assertion

Ref	Expression
2rexreu	⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 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 2rexreu

Step	Hyp	Ref	Expression
1		reurmo 3374	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
2		reurex 3373	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 𝜑)
3	2	rmoimi 42002	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
4	1, 3	syl 17	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
5		2reurex 42007	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
6	4, 5	anim12ci 609	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
7		reu5 3372	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
8	6, 7	sylibr 226	1 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∧ wa 386  ∃wrex 3119  ∃!wreu 3120  ∃*wrmo 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126

This theorem is referenced by:  2reu1  42012  2reu2  42013  2reu3  42014