Theorem 19.3 2187

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2190 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1977 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 2168	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2180	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 208	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1531  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778

This theorem is referenced by:  19.16  2210  19.17  2211  19.27  2212  19.28  2213  19.37  2217  aaan  2319  axrep4  5281  zfcndrep  10606  bj-alexbiex  36068  bj-alalbial  36070  fvineqsneq  36784