Theorem 2sb8e 2467

Description: An equivalent expression for double existence. (Contributed by Wolf Lammen, 2-Nov-2019.)

Assertion

Ref	Expression
2sb8e	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Distinct variable group:   𝑧,𝑤,𝜑

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2sb8e

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 ⊢ Ⅎ𝑤𝜑
2	1	sb8e 2425	. . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
3	2	exbii 1774	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤[𝑤 / 𝑦]𝜑)
4		excom 2042	. . 3 ⊢ (∃𝑥∃𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑)
5	3, 4	bitri 264	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑)
6		nfv 1843	. . . . 5 ⊢ Ⅎ𝑧𝜑
7	6	nfsb 2440	. . . 4 ⊢ Ⅎ𝑧[𝑤 / 𝑦]𝜑
8	7	sb8e 2425	. . 3 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9	8	exbii 1774	. 2 ⊢ (∃𝑤∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10		excom 2042	. 2 ⊢ (∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
11	5, 9, 10	3bitri 286	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 196  ∃wex 1704  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  2exsb  2469  2mo  2551