Theorem 19.41vvv 965

Description: Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers.

Assertion

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

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

Proof of Theorem 19.41vvv

Step	Hyp	Ref	Expression
1		19.41vv 964	. . 3 ⊢ (∃y∃z(φ ∧ ψ) ↔ (∃y∃zφ ∧ ψ))
2	1	biex 733	. 2 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ ∃x(∃y∃zφ ∧ ψ))
3		19.41v 963	. 2 ⊢ (∃x(∃y∃zφ ∧ ψ) ↔ (∃x∃y∃zφ ∧ ψ))
4	2, 3	bitr 151	1 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (∃x∃y∃zφ ∧ ψ))

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

This theorem is referenced by:  eloprabg 3035

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-gen 677  ax-17 925

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

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