Theorem r19.26-3 2749

Description: Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)

Assertion

Ref	Expression
r19.26-3	⊢ (∀x ∈ A (φ ∧ ψ ∧ χ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ ∧ ∀x ∈ A χ))

Proof of Theorem r19.26-3

Step	Hyp	Ref	Expression
1		df-3an 936	. . 3 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
2	1	ralbii 2639	. 2 ⊢ (∀x ∈ A (φ ∧ ψ ∧ χ) ↔ ∀x ∈ A ((φ ∧ ψ) ∧ χ))
3		r19.26 2747	. 2 ⊢ (∀x ∈ A ((φ ∧ ψ) ∧ χ) ↔ (∀x ∈ A (φ ∧ ψ) ∧ ∀x ∈ A χ))
4		r19.26 2747	. . . 4 ⊢ (∀x ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ))
5	4	anbi1i 676	. . 3 ⊢ ((∀x ∈ A (φ ∧ ψ) ∧ ∀x ∈ A χ) ↔ ((∀x ∈ A φ ∧ ∀x ∈ A ψ) ∧ ∀x ∈ A χ))
6		df-3an 936	. . 3 ⊢ ((∀x ∈ A φ ∧ ∀x ∈ A ψ ∧ ∀x ∈ A χ) ↔ ((∀x ∈ A φ ∧ ∀x ∈ A ψ) ∧ ∀x ∈ A χ))
7	5, 6	bitr4i 243	. 2 ⊢ ((∀x ∈ A (φ ∧ ψ) ∧ ∀x ∈ A χ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ ∧ ∀x ∈ A χ))
8	2, 3, 7	3bitri 262	1 ⊢ (∀x ∈ A (φ ∧ ψ ∧ χ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ ∧ ∀x ∈ A χ))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wral 2615

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-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by: (None)