Theorem 19.3 2200

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2203 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1986 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 2180	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2193	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786

This theorem is referenced by:  19.16  2225  19.17  2226  19.27  2227  19.28  2228  19.37  2232  axrep4  5159  zfcndrep  10025  bj-alexbiex  34146  bj-alalbial  34148  fvineqsneq  34829