Theorem ralcom4 3268

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 1939	. . . 4 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑦𝜑))
2	1	albii 1819	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑))
3		alcom 2159	. . 3 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3052	. . 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 3052	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
7	6	albii 1819	. 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 1538   ∈ wcel 2108  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-11 2157

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-ral 3052

This theorem is referenced by:  moelOLD  3384  ralxpxfr2d  3625  uniiunlem  4062  iunssf  5020  iunss  5021  disjor  5101  reliun  5795  idrefALT  6100  funimass4  6943  fnssintima  7355  ralrnmpo  7546  imaeqalov  7646  ralxp3f  8136  findcard3  9290  findcard3OLD  9291  ttrclss  9734  kmlem12  10176  fimaxre3  12188  vdwmc2  16999  ramtlecl  17020  iunocv  21641  1stccn  23401  itg2leub  25687  eqscut2  27770  addsuniflem  27960  mulsuniflem  28104  mptelee  28874  nmoubi  30753  nmopub  31889  nmfnleub  31906  disjorf  32560  funcnv5mpt  32646  untuni  35726  elintfv  35782  heibor1lem  37833  ineleq  38372  inecmo  38373  pmapglbx  39788  ismnuprim  44318  setrec1lem2  49552