Theorem 19.3v 2031

Description: Version of 19.3 2186 with a disjoint variable condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 2030. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2055. (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 1869	. 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
2		19.9v 2030	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑)
3	2	con2bii 349	. 2 ⊢ (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	1, 3	bitr4i 270	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 198  ∀wal 1599  ∃wex 1823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021

This theorem depends on definitions:  df-bi 199  df-ex 1824

This theorem is referenced by:  spvw  2032  19.27v  2038  19.28v  2039  19.37v  2040  axrep1  5007  kmlem14  9320  zfcndrep  9771  zfcndpow  9773  zfcndac  9776  bj-axrep1  33365  bj-snsetex  33523  iooelexlt  33805  dford4  38555  relexp0eq  38950