Theorem 3exdistr 970

Description: Distribution of existential quantifiers.

Assertion

Ref	Expression
3exdistr	⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃zχ)))

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

Proof of Theorem 3exdistr

Step	Hyp	Ref	Expression
1		3anass 585	. . . . . 6 ⊢ ((φ ∧ ψ ∧ χ) ↔ (φ ∧ (ψ ∧ χ)))
2	1	biex 733	. . . . 5 ⊢ (∃z(φ ∧ ψ ∧ χ) ↔ ∃z(φ ∧ (ψ ∧ χ)))
3		19.42v 966	. . . . 5 ⊢ (∃z(φ ∧ (ψ ∧ χ)) ↔ (φ ∧ ∃z(ψ ∧ χ)))
4		19.42v 966	. . . . . 6 ⊢ (∃z(ψ ∧ χ) ↔ (ψ ∧ ∃zχ))
5	4	anbi2i 367	. . . . 5 ⊢ ((φ ∧ ∃z(ψ ∧ χ)) ↔ (φ ∧ (ψ ∧ ∃zχ)))
6	2, 3, 5	3bitr 155	. . . 4 ⊢ (∃z(φ ∧ ψ ∧ χ) ↔ (φ ∧ (ψ ∧ ∃zχ)))
7	6	biex 733	. . 3 ⊢ (∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃y(φ ∧ (ψ ∧ ∃zχ)))
8		19.42v 966	. . 3 ⊢ (∃y(φ ∧ (ψ ∧ ∃zχ)) ↔ (φ ∧ ∃y(ψ ∧ ∃zχ)))
9	7, 8	bitr 151	. 2 ⊢ (∃y∃z(φ ∧ ψ ∧ χ) ↔ (φ ∧ ∃y(ψ ∧ ∃zχ)))
10	9	biex 733	1 ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃zχ)))

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

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-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

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