Theorem 19.3 2236

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2239 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 2001 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 2217	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2229	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 211	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1557  Ⅎwnf 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-ex 1799  df-nf 1803

This theorem is referenced by:  19.16  2259  19.17  2260  19.27  2261  19.28  2262  19.37  2266  aaan  2363  axrep4  5230  axrep4OLD  5231  zfcndrep  10566  bj-alexbiex  37135  bj-alalbial  37137  fvineqsneq  37867