Theorem 19.31 1876

Description: Theorem 19.31 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.31.1	⊢ Ⅎxψ

Assertion

Ref	Expression
19.31	⊢ (∀x(φ ∨ ψ) ↔ (∀xφ ∨ ψ))

Proof of Theorem 19.31

Step	Hyp	Ref	Expression
1		19.31.1	. . 3 ⊢ Ⅎxψ
2	1	19.32 1875	. 2 ⊢ (∀x(ψ ∨ φ) ↔ (ψ ∨ ∀xφ))
3		orcom 376	. . 3 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ))
4	3	albii 1566	. 2 ⊢ (∀x(φ ∨ ψ) ↔ ∀x(ψ ∨ φ))
5		orcom 376	. 2 ⊢ ((∀xφ ∨ ψ) ↔ (ψ ∨ ∀xφ))
6	2, 4, 5	3bitr4i 268	1 ⊢ (∀x(φ ∨ ψ) ↔ (∀xφ ∨ ψ))

Syntax hints:   ↔ wb 176   ∨ wo 357  ∀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-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  2eu3  2286