Theorem 19.3 2244

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2247 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 2009 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 2225	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2237	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1565  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-ex 1807  df-nf 1811

This theorem is referenced by:  19.16  2267  19.17  2268  19.27  2269  19.28  2270  19.37  2274  aaan  2371  axrep4  5245  axrep4OLD  5246  zfcndrep  10595  bj-alexbiex  37209  bj-alalbial  37211  fvineqsneq  37941