MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2sb8e Structured version   Visualization version   GIF version

Theorem 2sb8e 2534
Description: An equivalent expression for double existence. Usage of this theorem is discouraged because it depends on ax-13 2371. For a version requiring more disjoint variables, but fewer axioms, see 2sb8ef 2353. (Contributed by Wolf Lammen, 2-Nov-2019.) (New usage is discouraged.)
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 1918 . . . . 5 𝑤𝜑
21sb8e 2521 . . . 4 (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑)
32exbii 1851 . . 3 (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑤[𝑤 / 𝑦]𝜑)
4 excom 2163 . . 3 (∃𝑥𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
53, 4bitri 275 . 2 (∃𝑥𝑦𝜑 ↔ ∃𝑤𝑥[𝑤 / 𝑦]𝜑)
6 nfv 1918 . . . . 5 𝑧𝜑
76nfsb 2526 . . . 4 𝑧[𝑤 / 𝑦]𝜑
87sb8e 2521 . . 3 (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
98exbii 1851 . 2 (∃𝑤𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10 excom 2163 . 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 205  wex 1782  [wsb 2068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator