Theorem ralcom4 2878

Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

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		ralcom 2772	. 2 ⊢ (∀x ∈ A ∀y ∈ V φ ↔ ∀y ∈ V ∀x ∈ A φ)
2		ralv 2873	. . 3 ⊢ (∀y ∈ V φ ↔ ∀yφ)
3	2	ralbii 2639	. 2 ⊢ (∀x ∈ A ∀y ∈ V φ ↔ ∀x ∈ A ∀yφ)
4		ralv 2873	. 2 ⊢ (∀y ∈ V ∀x ∈ A φ ↔ ∀y∀x ∈ A φ)
5	1, 3, 4	3bitr3i 266	1 ⊢ (∀x ∈ A ∀yφ ↔ ∀y∀x ∈ A φ)

Syntax hints:   ↔ wb 176  ∀wal 1540  ∀wral 2615  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862

This theorem is referenced by:  uniiunlem  3354  iunss  4008  nnadjoinpw  4522  funimass4  5369  clos1induct  5881  dfnnc3  5886