Theorem 19.3v 2001

Description: Version of 19.3 2236 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 2003. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2027. (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 2000	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1929	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 211	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986

This theorem depends on definitions:  df-bi 209  df-ex 1799

This theorem is referenced by:  19.27v  2014  19.28v  2015  19.37v  2016  axrep1  5227  axrep4v  5231  axrep6OLD  5236  kmlem14  10117  zfcndrep  10569  zfcndpow  10571  zfcndac  10574  lfuhgr3  35434  bj-snsetex  37412  iooelexlt  37820  dford4  43570  relexp0eq  44241