Theorem 19.28 1870

Description: Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.28.1	⊢ Ⅎxφ

Assertion

Ref	Expression
19.28	⊢ (∀x(φ ∧ ψ) ↔ (φ ∧ ∀xψ))

Proof of Theorem 19.28

Step	Hyp	Ref	Expression
1		19.26 1593	. 2 ⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ∀xψ))
2		19.28.1	. . . 4 ⊢ Ⅎxφ
3	2	19.3 1785	. . 3 ⊢ (∀xφ ↔ φ)
4	3	anbi1i 676	. 2 ⊢ ((∀xφ ∧ ∀xψ) ↔ (φ ∧ ∀xψ))
5	1, 4	bitri 240	1 ⊢ (∀x(φ ∧ ψ) ↔ (φ ∧ ∀xψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544

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-ex 1542  df-nf 1545

This theorem is referenced by:  nfan1  1881  exan  1882  aaan  1884  19.28v  1895