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Theorem 2sb8e 2559
Description: An equivalent expression for double existence. (Contributed by Wolf Lammen, 2-Nov-2019.)
Assertion
Ref Expression
2sb8e (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Distinct variable group:   𝑧,𝑤,𝜑
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2sb8e
StepHypRef Expression
1 nfv 2009 . . . . 5 𝑤𝜑
21sb8e 2516 . . . 4 (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
32exbii 1943 . . 3 (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑤[𝑤 / 𝑦]𝜑)
4 excom 2206 . . 3 (∃𝑥𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
53, 4bitri 266 . 2 (∃𝑥𝑦𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
6 nfv 2009 . . . . 5 𝑧𝜑
76nfsb 2534 . . . 4 𝑧[𝑤 / 𝑦]𝜑
87sb8e 2516 . . 3 (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
98exbii 1943 . 2 (∃𝑤𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10 excom 2206 . 2 (∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
115, 9, 103bitri 288 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wex 1874  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063
This theorem is referenced by:  2exsb  2562  2mo  2673
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