Theorem ee4anv 982

Description: Rearrange existential quantifiers.

Assertion

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

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

Proof of Theorem ee4anv

Step	Hyp	Ref	Expression
1		excom 728	. . 3 ⊢ (∃y∃z∃w(φ ∧ ψ) ↔ ∃z∃y∃w(φ ∧ ψ))
2	1	biex 733	. 2 ⊢ (∃x∃y∃z∃w(φ ∧ ψ) ↔ ∃x∃z∃y∃w(φ ∧ ψ))
3		eeanv 980	. . 3 ⊢ (∃y∃w(φ ∧ ψ) ↔ (∃yφ ∧ ∃wψ))
4	3	bi2ex 734	. 2 ⊢ (∃x∃z∃y∃w(φ ∧ ψ) ↔ ∃x∃z(∃yφ ∧ ∃wψ))
5		eeanv 980	. 2 ⊢ (∃x∃z(∃yφ ∧ ∃wψ) ↔ (∃x∃yφ ∧ ∃z∃wψ))
6	2, 4, 5	3bitr 155	1 ⊢ (∃x∃y∃z∃w(φ ∧ ψ) ↔ (∃x∃yφ ∧ ∃z∃wψ))

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

This theorem is referenced by:  cgsex4g 1369  th3qlem1 3250  distrlem5pr 3925  5oalem7 5550  3oalem3 5554

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-7 676  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

metamath.org