Theorem 2reu7 48181

Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2682. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Assertion

Ref	Expression
2reu7	⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))

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

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

Proof of Theorem 2reu7

Step	Hyp	Ref	Expression
1		nfcv 2922	. . . 4 ⊢ Ⅎ𝑥𝐵
2		nfre1 3287	. . . 4 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑
3	1, 2	nfreuw 3395	. . 3 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑
4	3	reuan 3843	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
5		ancom 466	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
6	5	reubii 3374	. . . 4 ⊢ (∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑦 ∈ 𝐵 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
7		nfre1 3287	. . . . 5 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑
8	7	reuan 3843	. . . 4 ⊢ (∃!𝑦 ∈ 𝐵 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
9		ancom 466	. . . 4 ⊢ ((∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
10	6, 8, 9	3bitri 300	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
11	10	reubii 3374	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
12		ancom 466	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
13	4, 11, 12	3bitr4ri 307	1 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wrex 3086  ∃!wreu 3363

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:  2reu8  48182