Theorem 19.3 1785

Description: A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Hypothesis

Ref	Expression
19.3.1	⊢ Ⅎxφ

Assertion

Ref	Expression
19.3	⊢ (∀xφ ↔ φ)

Proof of Theorem 19.3

Step	Hyp	Ref	Expression
1		sp 1747	. 2 ⊢ (∀xφ → φ)
2		19.3.1	. . 3 ⊢ Ⅎxφ
3	2	nfri 1762	. 2 ⊢ (φ → ∀xφ)
4	1, 3	impbii 180	1 ⊢ (∀xφ ↔ φ)

Syntax hints:   ↔ 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:  19.16  1860  19.17  1861  19.27  1869  19.28  1870  19.37  1873  equsal  1960  2eu4  2287