Theorem 19.3 2202

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2205 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1981 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 2183	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2195	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 209	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  19.16  2225  19.17  2226  19.27  2227  19.28  2228  19.37  2232  aaan  2333  axrep4  5285  axrep4OLD  5286  zfcndrep  10654  bj-alexbiex  36700  bj-alalbial  36702  fvineqsneq  37413