Theorem ralcom4 1826

Description: Commutation of restricted and unrestricted universal quantifiers.

Assertion

Ref	Expression
ralcom4	⊢ (∀x ∈ A ∀yφ ↔ ∀y∀x ∈ A φ)

Distinct variable groups:   x,y   y,A

Proof of Theorem ralcom4

Step	Hyp	Ref	Expression
1		ralcom 1777	. 2 ⊢ (∀y ∈ V ∀x ∈ A φ ↔ ∀x ∈ A ∀y ∈ V φ)
2		ralv 1823	. 2 ⊢ (∀y ∈ V ∀x ∈ A φ ↔ ∀y∀x ∈ A φ)
3		ralv 1823	. . 3 ⊢ (∀y ∈ V φ ↔ ∀yφ)
4	3	ralbii 1670	. 2 ⊢ (∀x ∈ A ∀y ∈ V φ ↔ ∀x ∈ A ∀yφ)
5	1, 2, 4	3bitr3r 182	1 ⊢ (∀x ∈ A ∀yφ ↔ ∀y∀x ∈ A φ)

Syntax hints:   ↔ wb 146  ∀wal 956  ∀wral 1648  Vcvv 1814

This theorem is referenced by:  sbcralt 1993  sbcralgf 1995  reluni 3271  funimass4 3769  kmlem12 4786  ntreq0 7705  metcn4 7968

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815

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