Theorem 19.3v 1991

Description: Version of 19.3 2204 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 1993. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2020. (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 1990	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1917	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975

This theorem depends on definitions:  df-bi 210  df-ex 1787

This theorem is referenced by:  spvwOLD  1995  19.27v  2001  19.28v  2002  19.37v  2003  axrep1  5165  axrep6  5171  kmlem14  9675  zfcndrep  10126  zfcndpow  10128  zfcndac  10131  lfuhgr3  32664  bj-snsetex  34808  iooelexlt  35188  dford4  40463  relexp0eq  40895