Theorem 2sb8ef 2385

Description: An equivalent expression for double existence. Version of 2sb8e 2559 with more disjoint variable conditions, not requiring ax-13 2401. (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 2384	. . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
3	2	exbii 1881	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤[𝑤 / 𝑦]𝜑)
4		excom 2199	. . 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 2360	. . . 4 ⊢ Ⅎ𝑧[𝑤 / 𝑦]𝜑
8	7	sb8ef 2384	. . 3 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9	8	exbii 1881	. 2 ⊢ (∃𝑤∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10		excom 2199	. 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)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812  Ⅎwnf 1816  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  2exsb  2389  2mo  2673