Theorem 19.3v 2014

Description: Version of 19.3 2238 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 2016. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2040. (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 2013	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1942	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  19.27v  2027  19.28v  2028  19.37v  2029  axrep1  5232  axrep4v  5236  kmlem14  10245  zfcndrep  10702  zfcndpow  10704  zfcndac  10707  lfuhgr3  29733  bj-snsetex  37968  iooelexlt  38378  dford4  44129  relexp0eq  44794