Theorem 19.3 1564

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 1521	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfri 1529	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 126	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 105  ∀wal 1361  Ⅎwnf 1470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-4 1520

This theorem depends on definitions:  df-bi 117  df-nf 1471

This theorem is referenced by:  19.16  1565  19.17  1566  19.27  1571  19.28  1573  19.37-1  1684  rexxfrd  4477