Theorem 19.3v 1980

Description: Version of 19.3 2201 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 1982. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2006. (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 1979	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1909	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 209	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966

This theorem depends on definitions:  df-bi 207  df-ex 1779

This theorem is referenced by:  19.27v  1988  19.28v  1989  19.37v  1990  axrep1  5260  axrep4v  5264  axrep6OLD  5269  kmlem14  10186  zfcndrep  10636  zfcndpow  10638  zfcndac  10641  lfuhgr3  35084  bj-snsetex  36923  iooelexlt  37322  dford4  43004  relexp0eq  43676