Theorem 2exsb 2021

Description: An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)

Assertion

Ref	Expression
2exsb	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝑤,𝑧   𝜑,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2exsb

Step	Hyp	Ref	Expression
1		exsb 2020	. . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤∀𝑦(𝑦 = 𝑤 → 𝜑))
2	1	exbii 1616	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤∀𝑦(𝑦 = 𝑤 → 𝜑))
3		excom 1675	. . 3 ⊢ (∃𝑥∃𝑤∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑤∃𝑥∀𝑦(𝑦 = 𝑤 → 𝜑))
4	2, 3	bitri 184	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑤∃𝑥∀𝑦(𝑦 = 𝑤 → 𝜑))
5		exsb 2020	. . . 4 ⊢ (∃𝑥∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)))
6		impexp 263	. . . . . . . 8 ⊢ (((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)))
7	6	albii 1481	. . . . . . 7 ⊢ (∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)))
8		19.21v 1884	. . . . . . 7 ⊢ (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)))
9	7, 8	bitr2i 185	. . . . . 6 ⊢ ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
10	9	albii 1481	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
11	10	exbii 1616	. . . 4 ⊢ (∃𝑧∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∃𝑧∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
12	5, 11	bitri 184	. . 3 ⊢ (∃𝑥∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
13	12	exbii 1616	. 2 ⊢ (∃𝑤∃𝑥∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑤∃𝑧∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
14		excom 1675	. 2 ⊢ (∃𝑤∃𝑧∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
15	4, 13, 14	3bitri 206	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362  ∃wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by: (None)