New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  3exdistr GIF version

Theorem 3exdistr 1910
 Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
3exdistr (xyz(φ ψ χ) ↔ x(φ y(ψ zχ)))
Distinct variable groups:   φ,y   φ,z   ψ,z
Allowed substitution hints:   φ(x)   ψ(x,y)   χ(x,y,z)

Proof of Theorem 3exdistr
StepHypRef Expression
1 3anass 938 . . . 4 ((φ ψ χ) ↔ (φ (ψ χ)))
212exbii 1583 . . 3 (yz(φ ψ χ) ↔ yz(φ (ψ χ)))
3 19.42vv 1907 . . 3 (yz(φ (ψ χ)) ↔ (φ yz(ψ χ)))
4 exdistr 1906 . . . 4 (yz(ψ χ) ↔ y(ψ zχ))
54anbi2i 675 . . 3 ((φ yz(ψ χ)) ↔ (φ y(ψ zχ)))
62, 3, 53bitri 262 . 2 (yz(φ ψ χ) ↔ (φ y(ψ zχ)))
76exbii 1582 1 (xyz(φ ψ χ) ↔ x(φ y(ψ zχ)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator