Theorem 2sb8ef 2394

Description: An equivalent expression for double existence. Version of 2sb8e 2568 with more disjoint variable conditions, not requiring ax-13 2410. (Contributed by Wolf Lammen, 28-Jan-2023.)

Hypotheses

Ref	Expression
2sb8ef.1	⊢ Ⅎ𝑤𝜑
2sb8ef.2	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
2sb8ef	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb8ef

Step	Hyp	Ref	Expression
1		2sb8ef.1	. . . . 5 ⊢ Ⅎ𝑤𝜑
2	1	sb8ef 2393	. . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
3	2	exbii 1875	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤[𝑤 / 𝑦]𝜑)
4		excom 2203	. . 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		2sb8ef.2	. . . . 5 ⊢ Ⅎ𝑧𝜑
7	6	nfsbv 2369	. . . 4 ⊢ Ⅎ𝑧[𝑤 / 𝑦]𝜑
8	7	sb8ef 2393	. . 3 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9	8	exbii 1875	. 2 ⊢ (∃𝑤∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10		excom 2203	. 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 1806  Ⅎwnf 1810  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098

This theorem is referenced by:  2exsb  2398  2mo  2682