Theorem 19.41vvv 1903

Description: Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vvv	⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (∃x∃y∃zφ ∧ ψ))

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

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

Proof of Theorem 19.41vvv

Step	Hyp	Ref	Expression
1		19.41vv 1902	. . 3 ⊢ (∃y∃z(φ ∧ ψ) ↔ (∃y∃zφ ∧ ψ))
2	1	exbii 1582	. 2 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ ∃x(∃y∃zφ ∧ ψ))
3		19.41v 1901	. 2 ⊢ (∃x(∃y∃zφ ∧ ψ) ↔ (∃x∃y∃zφ ∧ ψ))
4	2, 3	bitri 240	1 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (∃x∃y∃zφ ∧ ψ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is used by:  19.41vvvv  1904  eloprabga  5579