Theorem 19.3 2131

Description: A wff may be quantified with a variable not free in it. Version of 19.9 2134 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1939 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 2111	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nf5ri 2123	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	impbii 201	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 198  ∀wal 1505  Ⅎwnf 1746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-ex 1743  df-nf 1747

This theorem is referenced by:  19.16  2157  19.17  2158  19.27  2159  19.28  2160  19.37  2164  axrep4  5054  zfcndrep  9834  bj-alexbiex  33549  bj-alalbial  33551  bj-axrep4  33627  fvineqsneq  34140