Theorem ralcom4 3147

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.)

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 1947	. . . . 5 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑦𝜑))
2	1	bicomi 227	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦𝜑) ↔ ∀𝑦(𝑥 ∈ 𝐴 → 𝜑))
3	2	albii 1827	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑))
4		alcom 2162	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5	3, 4	bitri 278	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑) ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-ral 3056	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑))
7		df-ral 3056	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	7	albii 1827	. 2 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
9	5, 6, 8	3bitr4i 306	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541   ∈ wcel 2112  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-11 2160

This theorem depends on definitions:  df-bi 210  df-ex 1788  df-ral 3056

This theorem is referenced by:  moel  3325  ralxpxfr2d  3543  uniiunlem  3985  iunssf  4939  iunss  4940  disjor  5019  reliun  5671  idrefALT  5958  funimass4  6755  ralrnmpo  7326  findcard3  8892  kmlem12  9740  fimaxre3  11743  vdwmc2  16495  ramtlecl  16516  iunocv  20597  1stccn  22314  itg2leub  24586  mptelee  26940  nmoubi  28807  nmopub  29943  nmfnleub  29960  disjorf  30591  funcnv5mpt  30679  untuni  33317  fnssintima  33345  ralxp3f  33355  elintfv  33408  eqscut2  33686  heibor1lem  35653  ineleq  36172  inecmo  36173  pmapglbx  37469  ss2iundf  40885  ismnuprim  41526  setrec1lem2  46008