Theorem ralcom4 3284

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 1937	. . . 4 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑦𝜑))
2	1	albii 1816	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑))
3		alcom 2157	. . 3 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3060	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑))
5	2, 3, 4	3bitr4ri 304	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-ral 3060	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
7	6	albii 1816	. 2 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	5, 7	bitr4i 278	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   ∈ wcel 2106  ∀wral 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-11 2155

This theorem depends on definitions:  df-bi 207  df-ex 1777  df-ral 3060

This theorem is referenced by:  moelOLD  3403  ralxpxfr2d  3646  uniiunlem  4097  iunssf  5049  iunss  5050  disjor  5130  reliun  5829  idrefALT  6134  funimass4  6973  fnssintima  7382  ralrnmpo  7572  imaeqalov  7672  ralxp3f  8161  findcard3  9316  findcard3OLD  9317  ttrclss  9758  kmlem12  10200  fimaxre3  12212  vdwmc2  17013  ramtlecl  17034  iunocv  21717  1stccn  23487  itg2leub  25784  eqscut2  27866  addsuniflem  28049  mulsuniflem  28190  mptelee  28925  nmoubi  30801  nmopub  31937  nmfnleub  31954  disjorf  32599  funcnv5mpt  32685  untuni  35689  elintfv  35746  heibor1lem  37796  ineleq  38336  inecmo  38337  pmapglbx  39752  ismnuprim  44290  setrec1lem2  48919