Theorem ralcom4 3291

Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)

Assertion

Ref	Expression
ralcom4	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem ralcom4

Step	Hyp	Ref	Expression
1		19.21v 1969	. . . 4 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑦𝜑))
2	1	albii 1849	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑))
3		alcom 2194	. . 3 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3080	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑))
5	2, 3, 4	3bitr4ri 307	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-ral 3080	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
7	6	albii 1849	. 2 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	5, 7	bitr4i 281	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568   ∈ wcel 2143  ∀wral 3079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-11 2192

This theorem depends on definitions:  df-bi 210  df-ex 1810  df-ral 3080

This theorem is referenced by:  ralxpxfr2d  3606  uniiunlem  4042  iunssf  5008  iunssfOLD  5009  iunss  5010  iunssOLD  5011  disjor  5092  replem  5250  idrefALT  6115  funimass4  6947  fnssintima  7362  ralrnmpo  7551  imaeqalov  7651  ralxp3f  8134  findcard3  9244  ttrclss  9690  kmlem12  10146  fimaxre3  12162  vdwmc2  17040  ramtlecl  17061  iunocv  21812  1stccn  23601  itg2leub  25874  eqcuts2  27960  addsuniflem  28175  mulsuniflem  28323  mpteleeOLD  29226  nmoubi  31105  nmopub  32241  nmfnleub  32258  disjorf  32905  funcnv5mpt  32993  untuni  36182  elintfv  36238  heibor1lem  38441  ineleq  38984  inecmo  38985  pmapglbx  40524  ismnuprim  44987  setrec1lem2  50449