Theorem 19.3v 2017

Description: Version of 19.3 2242 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 2019. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2043. (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 2016	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1945	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  19.27v  2030  19.28v  2031  19.37v  2032  axrep1  5237  axrep4v  5241  kmlem14  10272  zfcndrep  10729  zfcndpow  10731  zfcndac  10734  lfuhgr3  29761  bj-snsetex  38067  iooelexlt  38490  dford4  44240  relexp0eq  44904