Theorem 19.3v 1986

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 1988. (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 1985	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1911	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970

This theorem depends on definitions:  df-bi 210  df-ex 1782

This theorem is referenced by:  spvwOLD  1990  19.27v  1996  19.28v  1997  19.37v  1998  axrep1  5155  axrep6  5161  kmlem14  9574  zfcndrep  10025  zfcndpow  10027  zfcndac  10030  lfuhgr3  32479  bj-snsetex  34399  iooelexlt  34779  dford4  39970  relexp0eq  40402