Theorem 19.3v 1897

Description: Version of 19.3 2069 with a dv condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1896. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1935. (Revised by Wolf Lammen, 4-Dec-2017.)

Assertion

Ref	Expression
19.3v	⊢ (∀𝑥𝜑 ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem 19.3v

Step	Hyp	Ref	Expression
1		alex 1753	. 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
2		19.9v 1896	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑)
3	2	con2bii 347	. 2 ⊢ (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	1, 3	bitr4i 267	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1481  ∃wex 1704

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

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

This theorem is referenced by:  spvw  1898  19.27v  1908  19.28v  1909  19.37v  1910  axrep1  4772  kmlem14  8985  zfcndrep  9436  zfcndpow  9438  zfcndac  9441  bj-axrep1  32788  bj-snsetex  32951  iooelexlt  33210  dford4  37596  relexp0eq  37993