Theorem 2sb8e 2552

Description: An equivalent expression for double existence. Usage of this theorem is discouraged because it depends on ax-13 2379. For a version requiring more disjoint variables, but fewer axioms, see 2sb8ev 2364. (Contributed by Wolf Lammen, 2-Nov-2019.) (New usage is discouraged.)

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 1915	. . . . 5 ⊢ Ⅎ𝑤𝜑
2	1	sb8e 2537	. . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
3	2	exbii 1849	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤[𝑤 / 𝑦]𝜑)
4		excom 2166	. . 3 ⊢ (∃𝑥∃𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑)
5	3, 4	bitri 278	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑)
6		nfv 1915	. . . . 5 ⊢ Ⅎ𝑧𝜑
7	6	nfsb 2542	. . . 4 ⊢ Ⅎ𝑧[𝑤 / 𝑦]𝜑
8	7	sb8e 2537	. . 3 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9	8	exbii 1849	. 2 ⊢ (∃𝑤∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10		excom 2166	. 2 ⊢ (∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
11	5, 9, 10	3bitri 300	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1781  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070

This theorem is referenced by: (None)