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Theorem 4exdistr 1911
 Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
Assertion
Ref Expression
4exdistr (xyzw((φ ψ) (χ θ)) ↔ x(φ y(ψ z(χ wθ))))
Distinct variable groups:   φ,y   φ,z   φ,w   ψ,z   ψ,w   χ,w
Allowed substitution hints:   φ(x)   ψ(x,y)   χ(x,y,z)   θ(x,y,z,w)

Proof of Theorem 4exdistr
StepHypRef Expression
1 anass 630 . . . . . . . 8 (((φ ψ) (χ θ)) ↔ (φ (ψ (χ θ))))
21exbii 1582 . . . . . . 7 (w((φ ψ) (χ θ)) ↔ w(φ (ψ (χ θ))))
3 19.42v 1905 . . . . . . . 8 (w(φ (ψ (χ θ))) ↔ (φ w(ψ (χ θ))))
4 19.42v 1905 . . . . . . . . 9 (w(ψ (χ θ)) ↔ (ψ w(χ θ)))
54anbi2i 675 . . . . . . . 8 ((φ w(ψ (χ θ))) ↔ (φ (ψ w(χ θ))))
6 19.42v 1905 . . . . . . . . . 10 (w(χ θ) ↔ (χ wθ))
76anbi2i 675 . . . . . . . . 9 ((ψ w(χ θ)) ↔ (ψ (χ wθ)))
87anbi2i 675 . . . . . . . 8 ((φ (ψ w(χ θ))) ↔ (φ (ψ (χ wθ))))
93, 5, 83bitri 262 . . . . . . 7 (w(φ (ψ (χ θ))) ↔ (φ (ψ (χ wθ))))
102, 9bitri 240 . . . . . 6 (w((φ ψ) (χ θ)) ↔ (φ (ψ (χ wθ))))
1110exbii 1582 . . . . 5 (zw((φ ψ) (χ θ)) ↔ z(φ (ψ (χ wθ))))
12 19.42v 1905 . . . . 5 (z(φ (ψ (χ wθ))) ↔ (φ z(ψ (χ wθ))))
13 19.42v 1905 . . . . . 6 (z(ψ (χ wθ)) ↔ (ψ z(χ wθ)))
1413anbi2i 675 . . . . 5 ((φ z(ψ (χ wθ))) ↔ (φ (ψ z(χ wθ))))
1511, 12, 143bitri 262 . . . 4 (zw((φ ψ) (χ θ)) ↔ (φ (ψ z(χ wθ))))
1615exbii 1582 . . 3 (yzw((φ ψ) (χ θ)) ↔ y(φ (ψ z(χ wθ))))
17 19.42v 1905 . . 3 (y(φ (ψ z(χ wθ))) ↔ (φ y(ψ z(χ wθ))))
1816, 17bitri 240 . 2 (yzw((φ ψ) (χ θ)) ↔ (φ y(ψ z(χ wθ))))
1918exbii 1582 1 (xyzw((φ ψ) (χ θ)) ↔ x(φ y(ψ z(χ wθ))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃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-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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