Theorem 19.3 2203

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2206 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 2184	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2196	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 209	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1535  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-nf 1782

This theorem is referenced by:  19.16  2226  19.17  2227  19.27  2228  19.28  2229  19.37  2233  aaan  2337  axrep4  5308  zfcndrep  10683  bj-alexbiex  36665  bj-alalbial  36667  fvineqsneq  37378