ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2exsb GIF version

Theorem 2exsb 1933
Description: An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
Assertion
Ref Expression
2exsb (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝑤,𝑧   𝜑,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2exsb
StepHypRef Expression
1 exsb 1932 . . . 4 (∃𝑦𝜑 ↔ ∃𝑤𝑦(𝑦 = 𝑤𝜑))
21exbii 1541 . . 3 (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑤𝑦(𝑦 = 𝑤𝜑))
3 excom 1599 . . 3 (∃𝑥𝑤𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑))
42, 3bitri 182 . 2 (∃𝑥𝑦𝜑 ↔ ∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑))
5 exsb 1932 . . . 4 (∃𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
6 impexp 259 . . . . . . . 8 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76albii 1404 . . . . . . 7 (∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 19.21v 1801 . . . . . . 7 (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
97, 8bitr2i 183 . . . . . 6 ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
109albii 1404 . . . . 5 (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110exbii 1541 . . . 4 (∃𝑧𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∃𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
125, 11bitri 182 . . 3 (∃𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1312exbii 1541 . 2 (∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 excom 1599 . 2 (∃𝑤𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
154, 13, 143bitri 204 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator