Theorem ralcom4 3164

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 1946	. . . 4 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑦𝜑))
2	1	albii 1826	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦𝜑))
3		alcom 2160	. . 3 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3071	. . 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 3071	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
7	6	albii 1826	. 2 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	5, 7	bitr4i 277	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540   ∈ wcel 2110  ∀wral 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-11 2158

This theorem depends on definitions:  df-bi 206  df-ex 1787  df-ral 3071

This theorem is referenced by:  moelOLD  3358  ralxpxfr2d  3577  uniiunlem  4024  iunssf  4979  iunss  4980  disjor  5059  reliun  5725  idrefALT  6017  funimass4  6831  ralrnmpo  7407  findcard3  9045  ttrclss  9466  kmlem12  9928  fimaxre3  11932  vdwmc2  16691  ramtlecl  16712  iunocv  20897  1stccn  22625  itg2leub  24910  mptelee  27274  nmoubi  29143  nmopub  30279  nmfnleub  30296  disjorf  30927  funcnv5mpt  31014  untuni  33659  fnssintima  33685  ralxp3f  33694  elintfv  33747  eqscut2  34009  heibor1lem  35976  ineleq  36495  inecmo  36496  pmapglbx  37792  ss2iundf  41249  ismnuprim  41894  setrec1lem2  46373