Theorem 2sb8ef 2366

Description: An equivalent expression for double existence. Version of 2sb8e 2540 with more disjoint variable conditions, not requiring ax-13 2382. (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 2365	. . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
3	2	exbii 1856	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤[𝑤 / 𝑦]𝜑)
4		excom 2175	. . 3 ⊢ (∃𝑥∃𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑)
5	3, 4	bitri 277	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑)
6		2sb8ef.2	. . . . 5 ⊢ Ⅎ𝑧𝜑
7	6	nfsbv 2341	. . . 4 ⊢ Ⅎ𝑧[𝑤 / 𝑦]𝜑
8	7	sb8ef 2365	. . 3 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9	8	exbii 1856	. 2 ⊢ (∃𝑤∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10		excom 2175	. 2 ⊢ (∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
11	5, 9, 10	3bitri 299	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 208  ∃wex 1787  Ⅎwnf 1791  [wsb 2074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-11 2170  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-ex 1788  df-nf 1792  df-sb 2075

This theorem is referenced by:  2exsb  2370  2mo  2654