Theorem 19.42vvv 1786

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 1785	. . 3 ⊢ (∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃y∃zψ))
2	1	exbii 1493	. 2 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ ∃x(φ ∧ ∃y∃zψ))
3		19.42v 1783	. 2 ⊢ (∃x(φ ∧ ∃y∃zψ) ↔ (φ ∧ ∃x∃y∃zψ))
4	2, 3	bitri 173	1 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃x∃y∃zψ))

Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ceqsex6v  2592