Theorem albi 1354

Description: Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
albi	⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ))

Proof of Theorem albi

Step	Hyp	Ref	Expression
1		bi1 111	. . 3 ⊢ ((φ ↔ ψ) → (φ → ψ))
2	1	al2imi 1344	. 2 ⊢ (∀x(φ ↔ ψ) → (∀xφ → ∀xψ))
3		bi2 121	. . 3 ⊢ ((φ ↔ ψ) → (ψ → φ))
4	3	al2imi 1344	. 2 ⊢ (∀x(φ ↔ ψ) → (∀xψ → ∀xφ))
5	2, 4	impbid 120	1 ⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  albii  1356  albidh  1366  19.16  1444  19.17  1445  intmin4  3634  dfiin2g  3681