Theorem 19.3v 1989

Description: Version of 19.3 2214 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 1991. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)

Assertion

Ref	Expression
19.3v	⊢ (∀𝑥𝜑 ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem 19.3v

Step	Hyp	Ref	Expression
1		spvw 1988	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1917	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 210	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 207  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974

This theorem depends on definitions:  df-bi 208  df-ex 1787

This theorem is referenced by:  19.27v  2002  19.28v  2003  19.37v  2004  axrep1  5207  axrep4v  5211  axrep6OLD  5216  kmlem14  10084  zfcndrep  10535  zfcndpow  10537  zfcndac  10540  lfuhgr3  35355  bj-snsetex  37323  iooelexlt  37731  dford4  43481  relexp0eq  44152