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	⊢ (∃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 938	. . . 4 ⊢ ((φ ∧ ψ ∧ χ) ↔ (φ ∧ (ψ ∧ χ)))
2	1	2exbii 1583	. . 3 ⊢ (∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃y∃z(φ ∧ (ψ ∧ χ)))
3		19.42vv 1907	. . 3 ⊢ (∃y∃z(φ ∧ (ψ ∧ χ)) ↔ (φ ∧ ∃y∃z(ψ ∧ χ)))
4		exdistr 1906	. . . 4 ⊢ (∃y∃z(ψ ∧ χ) ↔ ∃y(ψ ∧ ∃zχ))
5	4	anbi2i 675	. . 3 ⊢ ((φ ∧ ∃y∃z(ψ ∧ χ)) ↔ (φ ∧ ∃y(ψ ∧ ∃zχ)))
6	2, 3, 5	3bitri 262	. 2 ⊢ (∃y∃z(φ ∧ ψ ∧ χ) ↔ (φ ∧ ∃y(ψ ∧ ∃zχ)))
7	6	exbii 1582	1 ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃zχ)))

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)