Theorem 4exdistr 1911

Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Assertion

Ref	Expression
4exdistr	⊢ (∃x∃y∃z∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃z(χ ∧ ∃wθ))))

Distinct variable groups:   φ,y   φ,z   φ,w   ψ,z   ψ,w   χ,w

Allowed substitution hints:   φ(x)   ψ(x,y)   χ(x,y,z)   θ(x,y,z,w)

Proof of Theorem 4exdistr

Step	Hyp	Ref	Expression
1		anass 630	. . . . . . . 8 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ (φ ∧ (ψ ∧ (χ ∧ θ))))
2	1	exbii 1582	. . . . . . 7 ⊢ (∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ∃w(φ ∧ (ψ ∧ (χ ∧ θ))))
3		19.42v 1905	. . . . . . . 8 ⊢ (∃w(φ ∧ (ψ ∧ (χ ∧ θ))) ↔ (φ ∧ ∃w(ψ ∧ (χ ∧ θ))))
4		19.42v 1905	. . . . . . . . 9 ⊢ (∃w(ψ ∧ (χ ∧ θ)) ↔ (ψ ∧ ∃w(χ ∧ θ)))
5	4	anbi2i 675	. . . . . . . 8 ⊢ ((φ ∧ ∃w(ψ ∧ (χ ∧ θ))) ↔ (φ ∧ (ψ ∧ ∃w(χ ∧ θ))))
6		19.42v 1905	. . . . . . . . . 10 ⊢ (∃w(χ ∧ θ) ↔ (χ ∧ ∃wθ))
7	6	anbi2i 675	. . . . . . . . 9 ⊢ ((ψ ∧ ∃w(χ ∧ θ)) ↔ (ψ ∧ (χ ∧ ∃wθ)))
8	7	anbi2i 675	. . . . . . . 8 ⊢ ((φ ∧ (ψ ∧ ∃w(χ ∧ θ))) ↔ (φ ∧ (ψ ∧ (χ ∧ ∃wθ))))
9	3, 5, 8	3bitri 262	. . . . . . 7 ⊢ (∃w(φ ∧ (ψ ∧ (χ ∧ θ))) ↔ (φ ∧ (ψ ∧ (χ ∧ ∃wθ))))
10	2, 9	bitri 240	. . . . . 6 ⊢ (∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ (φ ∧ (ψ ∧ (χ ∧ ∃wθ))))
11	10	exbii 1582	. . . . 5 ⊢ (∃z∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ∃z(φ ∧ (ψ ∧ (χ ∧ ∃wθ))))
12		19.42v 1905	. . . . 5 ⊢ (∃z(φ ∧ (ψ ∧ (χ ∧ ∃wθ))) ↔ (φ ∧ ∃z(ψ ∧ (χ ∧ ∃wθ))))
13		19.42v 1905	. . . . . 6 ⊢ (∃z(ψ ∧ (χ ∧ ∃wθ)) ↔ (ψ ∧ ∃z(χ ∧ ∃wθ)))
14	13	anbi2i 675	. . . . 5 ⊢ ((φ ∧ ∃z(ψ ∧ (χ ∧ ∃wθ))) ↔ (φ ∧ (ψ ∧ ∃z(χ ∧ ∃wθ))))
15	11, 12, 14	3bitri 262	. . . 4 ⊢ (∃z∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ (φ ∧ (ψ ∧ ∃z(χ ∧ ∃wθ))))
16	15	exbii 1582	. . 3 ⊢ (∃y∃z∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ∃y(φ ∧ (ψ ∧ ∃z(χ ∧ ∃wθ))))
17		19.42v 1905	. . 3 ⊢ (∃y(φ ∧ (ψ ∧ ∃z(χ ∧ ∃wθ))) ↔ (φ ∧ ∃y(ψ ∧ ∃z(χ ∧ ∃wθ))))
18	16, 17	bitri 240	. 2 ⊢ (∃y∃z∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ (φ ∧ ∃y(ψ ∧ ∃z(χ ∧ ∃wθ))))
19	18	exbii 1582	1 ⊢ (∃x∃y∃z∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃z(χ ∧ ∃wθ))))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃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-ex 1542  df-nf 1545

This theorem is referenced by: (None)