Theorem exdistr2 1909

Description: Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Assertion

Ref	Expression
exdistr2	⊢ (∃x∃y∃z(φ ∧ ψ) ↔ ∃x(φ ∧ ∃y∃zψ))

Distinct variable groups:   φ,y   φ,z

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

Proof of Theorem exdistr2

Step	Hyp	Ref	Expression
1		19.42vv 1907	. 2 ⊢ (∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃y∃zψ))
2	1	exbii 1582	1 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ ∃x(φ ∧ ∃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: (None)