Theorem 19.3v 2012

Description: Version of 19.3 2238 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 2014. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2038. (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 2011	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1940	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997

This theorem depends on definitions:  df-bi 210  df-ex 1810

This theorem is referenced by:  19.27v  2025  19.28v  2026  19.37v  2027  axrep1  5240  axrep4v  5244  axrep6OLD  5249  kmlem14  10148  zfcndrep  10600  zfcndpow  10602  zfcndac  10605  lfuhgr3  35590  bj-snsetex  37577  iooelexlt  37986  dford4  43736  relexp0eq  44407