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Theorem 3exdistr 1789
 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 888 . . . 4 ((φ ψ χ) ↔ (φ (ψ χ)))
212exbii 1494 . . 3 (yz(φ ψ χ) ↔ yz(φ (ψ χ)))
3 19.42vv 1785 . . 3 (yz(φ (ψ χ)) ↔ (φ yz(ψ χ)))
4 exdistr 1784 . . . 4 (yz(ψ χ) ↔ y(ψ zχ))
54anbi2i 430 . . 3 ((φ yz(ψ χ)) ↔ (φ y(ψ zχ)))
62, 3, 53bitri 195 . 2 (yz(φ ψ χ) ↔ (φ y(ψ zχ)))
76exbii 1493 1 (xyz(φ ψ χ) ↔ x(φ y(ψ zχ)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 884  ∃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  df-3an 886 This theorem is referenced by: (None)
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