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	⊢ (∃x∃y∃z(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

Step	Hyp	Ref	Expression
1		3anass 888	. . . 4 ⊢ ((φ ∧ ψ ∧ χ) ↔ (φ ∧ (ψ ∧ χ)))
2	1	2exbii 1494	. . 3 ⊢ (∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃y∃z(φ ∧ (ψ ∧ χ)))
3		19.42vv 1785	. . 3 ⊢ (∃y∃z(φ ∧ (ψ ∧ χ)) ↔ (φ ∧ ∃y∃z(ψ ∧ χ)))
4		exdistr 1784	. . . 4 ⊢ (∃y∃z(ψ ∧ χ) ↔ ∃y(ψ ∧ ∃zχ))
5	4	anbi2i 430	. . 3 ⊢ ((φ ∧ ∃y∃z(ψ ∧ χ)) ↔ (φ ∧ ∃y(ψ ∧ ∃zχ)))
6	2, 3, 5	3bitri 195	. 2 ⊢ (∃y∃z(φ ∧ ψ ∧ χ) ↔ (φ ∧ ∃y(ψ ∧ ∃zχ)))
7	6	exbii 1493	1 ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃zχ)))

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)