Theorem 19.3 2214

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2217 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1989 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 2195	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2207	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 210	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 207  ∀wal 1545  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  19.16  2237  19.17  2238  19.27  2239  19.28  2240  19.37  2244  aaan  2341  axrep4  5205  axrep4OLD  5206  zfcndrep  10528  bj-alexbiex  37042  bj-alalbial  37044  fvineqsneq  37774