Theorem ralcom4 1360

Description: Commutation of restricted and unrestricted universal quantifiers.

Assertion

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

Distinct variable group(s):   x,y   y,A

Proof of Theorem ralcom4

Step	Hyp	Ref	Expression
1		ralcom 1312	. 2 ⊢ (∀y ∈ V ∀x ∈ A φ ↔ ∀x ∈ A ∀y ∈ V φ)
2		ralv 1357	. 2 ⊢ (∀y ∈ V ∀x ∈ A φ ↔ ∀y∀x ∈ A φ)
3		ralv 1357	. . 3 ⊢ (∀y ∈ V φ ↔ ∀yφ)
4	3	biral 1223	. 2 ⊢ (∀x ∈ A ∀y ∈ V φ ↔ ∀x ∈ A ∀yφ)
5	1, 2, 4	3bitr3r 157	1 ⊢ (∀x ∈ A ∀yφ ↔ ∀y∀x ∈ A φ)

Syntax hints:   ↔ wb 127  ∀wal 672  ∀wral 1201  Vcvv 1348

This theorem is referenced by:  reluni 2493  kmlem11 3590

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349

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