Theorem 19.42vvv 1908

Description: Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)

Assertion

Ref	Expression
19.42vvv	⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃x∃y∃zψ))

Distinct variable groups:   φ,x   φ,y   φ,z

Allowed substitution hints:   ψ(x,y,z)

Proof of Theorem 19.42vvv

Step	Hyp	Ref	Expression
1		19.42vv 1907	. . 3 ⊢ (∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃y∃zψ))
2	1	exbii 1582	. 2 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ ∃x(φ ∧ ∃y∃zψ))
3		19.42v 1905	. 2 ⊢ (∃x(φ ∧ ∃y∃zψ) ↔ (φ ∧ ∃x∃y∃zψ))
4	2, 3	bitri 240	1 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃x∃y∃zψ))

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:  ceqsex6v  2900