Theorem 19.16 1860

Description: Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.16.1	⊢ Ⅎxφ

Assertion

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

Proof of Theorem 19.16

Step	Hyp	Ref	Expression
1		19.16.1	. . 3 ⊢ Ⅎxφ
2	1	19.3 1785	. 2 ⊢ (∀xφ ↔ φ)
3		albi 1564	. 2 ⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ))
4	2, 3	syl5bbr 250	1 ⊢ (∀x(φ ↔ ψ) → (φ ↔ ∀xψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀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-ex 1542  df-nf 1545

This theorem is referenced by: (None)