Theorem eeeanv 981

Description: Rearrange existential quantifiers.

Assertion

Ref	Expression
eeeanv	⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ (∃xφ ∧ ∃yψ ∧ ∃zχ))

Distinct variable group(s):   y,z,φ   x,z,ψ   x,y,χ

Proof of Theorem eeeanv

Step	Hyp	Ref	Expression
1		19.42vv 968	. . . . 5 ⊢ (∃y∃z(φ ∧ (ψ ∧ χ)) ↔ (φ ∧ ∃y∃z(ψ ∧ χ)))
2		eeanv 980	. . . . . 6 ⊢ (∃y∃z(ψ ∧ χ) ↔ (∃yψ ∧ ∃zχ))
3	2	anbi2i 367	. . . . 5 ⊢ ((φ ∧ ∃y∃z(ψ ∧ χ)) ↔ (φ ∧ (∃yψ ∧ ∃zχ)))
4	1, 3	bitr 151	. . . 4 ⊢ (∃y∃z(φ ∧ (ψ ∧ χ)) ↔ (φ ∧ (∃yψ ∧ ∃zχ)))
5	4	biex 733	. . 3 ⊢ (∃x∃y∃z(φ ∧ (ψ ∧ χ)) ↔ ∃x(φ ∧ (∃yψ ∧ ∃zχ)))
6		19.41v 963	. . 3 ⊢ (∃x(φ ∧ (∃yψ ∧ ∃zχ)) ↔ (∃xφ ∧ (∃yψ ∧ ∃zχ)))
7	5, 6	bitr 151	. 2 ⊢ (∃x∃y∃z(φ ∧ (ψ ∧ χ)) ↔ (∃xφ ∧ (∃yψ ∧ ∃zχ)))
8		3anass 585	. . 3 ⊢ ((φ ∧ ψ ∧ χ) ↔ (φ ∧ (ψ ∧ χ)))
9	8	bi3ex 735	. 2 ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃x∃y∃z(φ ∧ (ψ ∧ χ)))
10		3anass 585	. 2 ⊢ ((∃xφ ∧ ∃yψ ∧ ∃zχ) ↔ (∃xφ ∧ (∃yψ ∧ ∃zχ)))
11	7, 9, 10	3bitr4 158	1 ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ (∃xφ ∧ ∃yψ ∧ ∃zχ))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∧ w3a 581  ∃wex 678

This theorem is referenced by:  vtocl3 1380  eloprabg 3035

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679

metamath.org