Theorem albi 1564

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 178	. . 3 ⊢ ((φ ↔ ψ) → (φ → ψ))
2	1	al2imi 1561	. 2 ⊢ (∀x(φ ↔ ψ) → (∀xφ → ∀xψ))
3		bi2 189	. . 3 ⊢ ((φ ↔ ψ) → (ψ → φ))
4	3	al2imi 1561	. 2 ⊢ (∀x(φ ↔ ψ) → (∀xψ → ∀xφ))
5	2, 4	impbid 183	1 ⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This proof depends on definitions:  df-bi 177

This theorem is used by:  albii  1566  albidh  1590  19.16  1860  19.17  1861  intmin4  3956  dfiin2g  4001