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Theorem 2sb8ev 2371
Description: An equivalent expression for double existence. Version of 2sb8e 2572 with more disjoint variable conditions, not requiring ax-13 2386. (Contributed by Wolf Lammen, 28-Jan-2023.)
Hypotheses
Ref Expression
2sb8ev.1 𝑤𝜑
2sb8ev.2 𝑧𝜑
Assertion
Ref Expression
2sb8ev (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝑧   𝑧,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb8ev
StepHypRef Expression
1 2sb8ev.1 . . . . 5 𝑤𝜑
21sb8ev 2370 . . . 4 (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
32exbii 1844 . . 3 (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑤[𝑤 / 𝑦]𝜑)
4 excom 2165 . . 3 (∃𝑥𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
53, 4bitri 277 . 2 (∃𝑥𝑦𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
6 2sb8ev.2 . . . . 5 𝑧𝜑
76nfsbv 2345 . . . 4 𝑧[𝑤 / 𝑦]𝜑
87sb8ev 2370 . . 3 (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
98exbii 1844 . 2 (∃𝑤𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10 excom 2165 . 2 (∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
115, 9, 103bitri 299 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wex 1776  wnf 1780  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066
This theorem is referenced by:  2exsb  2375  2mo  2729
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