Theorem 19.3 2191

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2194 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1978 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 2172	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2184	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 208	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1532  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-ex 1775  df-nf 1779

This theorem is referenced by:  19.16  2214  19.17  2215  19.27  2216  19.28  2217  19.37  2221  aaan  2323  axrep4  5284  zfcndrep  10631  bj-alexbiex  36170  bj-alalbial  36172  fvineqsneq  36885