Theorem 19.3v 1986

Description: Version of 19.3 2196 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 1988. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2012. (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 1985	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1914	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 208	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  19.27v  1994  19.28v  1995  19.37v  1996  axrep1  5285  axrep6  5291  kmlem14  10154  zfcndrep  10605  zfcndpow  10607  zfcndac  10610  lfuhgr3  34048  bj-snsetex  35782  iooelexlt  36181  dford4  41701  relexp0eq  42385