ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2sb5 GIF version

Theorem 2sb5 1936
Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
2sb5 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb5
StepHypRef Expression
1 sb5 1843 . 2 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑))
2 19.42v 1862 . . . 4 (∃𝑦(𝑥 = 𝑧 ∧ (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 ∧ ∃𝑦(𝑦 = 𝑤𝜑)))
3 anass 398 . . . . 5 (((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑) ↔ (𝑥 = 𝑧 ∧ (𝑦 = 𝑤𝜑)))
43exbii 1569 . . . 4 (∃𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝑧 ∧ (𝑦 = 𝑤𝜑)))
5 sb5 1843 . . . . 5 ([𝑤 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑤𝜑))
65anbi2i 452 . . . 4 ((𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑦(𝑦 = 𝑤𝜑)))
72, 4, 63bitr4ri 212 . . 3 ((𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
87exbii 1569 . 2 (∃𝑥(𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
91, 8bitri 183 1 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1453  [wsb 1720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-sb 1721
This theorem is referenced by:  opelopabsbALT  4151
  Copyright terms: Public domain W3C validator