Theorem 19.3v 1979

Description: Version of 19.3 2200 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 1981. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2005. (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 1978	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1908	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 209	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965

This theorem depends on definitions:  df-bi 207  df-ex 1777

This theorem is referenced by:  19.27v  1987  19.28v  1988  19.37v  1989  axrep1  5286  axrep4v  5290  axrep6OLD  5295  kmlem14  10202  zfcndrep  10652  zfcndpow  10654  zfcndac  10657  lfuhgr3  35104  bj-snsetex  36946  iooelexlt  37345  dford4  43018  relexp0eq  43691