Theorem 19.3 2242

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2245 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 2017 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 2223	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2235	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  19.16  2265  19.17  2266  19.27  2267  19.28  2268  19.37  2272  aaan  2366  axrep4  5244  zfcndrep  10738  srepdflem1  36126  bj-alexbiex  37816  bj-alalbial  37818  fvineqsneq  38603