Theorem 19.16 730

Description: Theorem 19.16 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.16.1	⊢ (φ → ∀xφ)

Assertion

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

Proof of Theorem 19.16

Step	Hyp	Ref	Expression
1		19.15 694	. 2 ⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ))
2		19.16.1	. . 3 ⊢ (φ → ∀xφ)
3	2	19.3r 714	. 2 ⊢ (φ ↔ ∀xφ)
4	1, 3	syl5bb 410	1 ⊢ (∀x(φ ↔ ψ) → (φ ↔ ∀xψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org