Theorem 19.26-3an 1595

Description: Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)

Assertion

Ref	Expression
19.26-3an	⊢ (∀x(φ ∧ ψ ∧ χ) ↔ (∀xφ ∧ ∀xψ ∧ ∀xχ))

Proof of Theorem 19.26-3an

Step	Hyp	Ref	Expression
1		19.26 1593	. . 3 ⊢ (∀x((φ ∧ ψ) ∧ χ) ↔ (∀x(φ ∧ ψ) ∧ ∀xχ))
2		19.26 1593	. . . 4 ⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ∀xψ))
3	2	anbi1i 676	. . 3 ⊢ ((∀x(φ ∧ ψ) ∧ ∀xχ) ↔ ((∀xφ ∧ ∀xψ) ∧ ∀xχ))
4	1, 3	bitri 240	. 2 ⊢ (∀x((φ ∧ ψ) ∧ χ) ↔ ((∀xφ ∧ ∀xψ) ∧ ∀xχ))
5		df-3an 936	. . 3 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
6	5	albii 1566	. 2 ⊢ (∀x(φ ∧ ψ ∧ χ) ↔ ∀x((φ ∧ ψ) ∧ χ))
7		df-3an 936	. 2 ⊢ ((∀xφ ∧ ∀xψ ∧ ∀xχ) ↔ ((∀xφ ∧ ∀xψ) ∧ ∀xχ))
8	4, 6, 7	3bitr4i 268	1 ⊢ (∀x(φ ∧ ψ ∧ χ) ↔ (∀xφ ∧ ∀xψ ∧ ∀xχ))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is referenced by: (None)