Theorem 19.3 2069

Description: A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1897 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 2053	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2065	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 199	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  19.16  2093  19.17  2094  19.27  2095  19.28  2096  19.37  2100  axrep4  4775  zfcndrep  9436  bj-alexbiex  32690  bj-alalbial  32692  bj-axrep4  32791