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Theorem 2sb8ef 2357
Description: An equivalent expression for double existence. Version of 2sb8e 2533 with more disjoint variable conditions, not requiring ax-13 2375. (Contributed by Wolf Lammen, 28-Jan-2023.)
Hypotheses
Ref Expression
2sb8ef.1 𝑤𝜑
2sb8ef.2 𝑧𝜑
Assertion
Ref Expression
2sb8ef (∃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 2sb8ef
StepHypRef Expression
1 2sb8ef.1 . . . . 5 𝑤𝜑
21sb8ef 2356 . . . 4 (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
32exbii 1845 . . 3 (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑤[𝑤 / 𝑦]𝜑)
4 excom 2160 . . 3 (∃𝑥𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
53, 4bitri 275 . 2 (∃𝑥𝑦𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
6 2sb8ef.2 . . . . 5 𝑧𝜑
76nfsbv 2329 . . . 4 𝑧[𝑤 / 𝑦]𝜑
87sb8ef 2356 . . 3 (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
98exbii 1845 . 2 (∃𝑤𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10 excom 2160 . 2 (∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
115, 9, 103bitri 297 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wex 1776  wnf 1780  [wsb 2062
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 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by:  2exsb  2361  2mo  2646
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