Theorem 19.3 2067

Description: A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1894 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-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 2051	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2063	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 199	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∀wal 1478  Ⅎwnf 1705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044

This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707

This theorem is referenced by:  19.16  2091  19.17  2092  19.27  2093  19.28  2094  19.37  2098  axrep4  4745  zfcndrep  9396  bj-alexbiex  32385  bj-alalbial  32387  bj-axrep4  32487