Theorem 2rexreu 3719

Description: Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2671. (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 3368	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
2		reurex 3369	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 𝜑)
3	2	rmoimi 3699	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
4	1, 3	syl 18	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
5		2reurex 3717	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
6	4, 5	anim12ci 626	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
7		reu5 3367	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
8	6, 7	sylibr 237	1 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wrex 3086  ∃!wreu 3363  ∃*wrmo 3364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-mo 2564  df-eu 2594  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366

This theorem is used by:  2reu1  3844  2reu2  3845  opreu2reu  6298  2reu2reu2  33126  2reu3  48273