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Theorem 4exdistr 1790
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 381 . . . . . . . 8 (((φ ψ) (χ θ)) ↔ (φ (ψ (χ θ))))
21exbii 1493 . . . . . . 7 (w((φ ψ) (χ θ)) ↔ w(φ (ψ (χ θ))))
3 19.42v 1783 . . . . . . . 8 (w(φ (ψ (χ θ))) ↔ (φ w(ψ (χ θ))))
4 19.42v 1783 . . . . . . . . 9 (w(ψ (χ θ)) ↔ (ψ w(χ θ)))
54anbi2i 430 . . . . . . . 8 ((φ w(ψ (χ θ))) ↔ (φ (ψ w(χ θ))))
6 19.42v 1783 . . . . . . . . . 10 (w(χ θ) ↔ (χ wθ))
76anbi2i 430 . . . . . . . . 9 ((ψ w(χ θ)) ↔ (ψ (χ wθ)))
87anbi2i 430 . . . . . . . 8 ((φ (ψ w(χ θ))) ↔ (φ (ψ (χ wθ))))
93, 5, 83bitri 195 . . . . . . 7 (w(φ (ψ (χ θ))) ↔ (φ (ψ (χ wθ))))
102, 9bitri 173 . . . . . 6 (w((φ ψ) (χ θ)) ↔ (φ (ψ (χ wθ))))
1110exbii 1493 . . . . 5 (zw((φ ψ) (χ θ)) ↔ z(φ (ψ (χ wθ))))
12 19.42v 1783 . . . . 5 (z(φ (ψ (χ wθ))) ↔ (φ z(ψ (χ wθ))))
13 19.42v 1783 . . . . . 6 (z(ψ (χ wθ)) ↔ (ψ z(χ wθ)))
1413anbi2i 430 . . . . 5 ((φ z(ψ (χ wθ))) ↔ (φ (ψ z(χ wθ))))
1511, 12, 143bitri 195 . . . 4 (zw((φ ψ) (χ θ)) ↔ (φ (ψ z(χ wθ))))
1615exbii 1493 . . 3 (yzw((φ ψ) (χ θ)) ↔ y(φ (ψ z(χ wθ))))
17 19.42v 1783 . . 3 (y(φ (ψ z(χ wθ))) ↔ (φ y(ψ z(χ wθ))))
1816, 17bitri 173 . 2 (yzw((φ ψ) (χ θ)) ↔ (φ y(ψ z(χ wθ))))
1918exbii 1493 1 (xyzw((φ ψ) (χ θ)) ↔ x(φ y(ψ z(χ wθ))))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424
This theorem depends on definitions:  df-bi 110
This theorem is referenced by: (None)
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