Theorem ee4anv 1915

Description: Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)

Assertion

Ref	Expression
ee4anv	⊢ (∃x∃y∃z∃w(φ ∧ ψ) ↔ (∃x∃yφ ∧ ∃z∃wψ))

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

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

Proof of Theorem ee4anv

Step	Hyp	Ref	Expression
1		excom 1741	. . 3 ⊢ (∃y∃z∃w(φ ∧ ψ) ↔ ∃z∃y∃w(φ ∧ ψ))
2	1	exbii 1582	. 2 ⊢ (∃x∃y∃z∃w(φ ∧ ψ) ↔ ∃x∃z∃y∃w(φ ∧ ψ))
3		eeanv 1913	. . 3 ⊢ (∃y∃w(φ ∧ ψ) ↔ (∃yφ ∧ ∃wψ))
4	3	2exbii 1583	. 2 ⊢ (∃x∃z∃y∃w(φ ∧ ψ) ↔ ∃x∃z(∃yφ ∧ ∃wψ))
5		eeanv 1913	. 2 ⊢ (∃x∃z(∃yφ ∧ ∃wψ) ↔ (∃x∃yφ ∧ ∃z∃wψ))
6	2, 4, 5	3bitri 262	1 ⊢ (∃x∃y∃z∃w(φ ∧ ψ) ↔ (∃x∃yφ ∧ ∃z∃wψ))

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-7 1734  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:  cgsex4g  2893  funsi  5521  fntxp  5805  fnpprod  5844