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Theorem 2sb8e 2495
 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 1883 . . . . 5 𝑤𝜑
21sb8e 2453 . . . 4 (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
32exbii 1814 . . 3 (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑤[𝑤 / 𝑦]𝜑)
4 excom 2082 . . 3 (∃𝑥𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
53, 4bitri 264 . 2 (∃𝑥𝑦𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
6 nfv 1883 . . . . 5 𝑧𝜑
76nfsb 2468 . . . 4 𝑧[𝑤 / 𝑦]𝜑
87sb8e 2453 . . 3 (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
98exbii 1814 . 2 (∃𝑤𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10 excom 2082 . 2 (∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
115, 9, 103bitri 286 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∃wex 1744  [wsb 1937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938 This theorem is referenced by:  2exsb  2497  2mo  2580
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