Theorem 19.3 2198

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2201 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 2178	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2191	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 208	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1537  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-nf 1788

This theorem is referenced by:  19.16  2221  19.17  2222  19.27  2223  19.28  2224  19.37  2228  axrep4  5210  zfcndrep  10301  bj-alexbiex  34808  bj-alalbial  34810  fvineqsneq  35510