ralcom4 - New Foundations Explorer

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Theorem ralcom4 2877

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 2771	. 2 ⊢ (∀x ∈ A ∀y ∈ V φ ↔ ∀y ∈ V ∀x ∈ A φ)
2		ralv 2872	. . 3 ⊢ (∀y ∈ V φ ↔ ∀yφ)
3	2	ralbii 2638	. 2 ⊢ (∀x ∈ A ∀y ∈ V φ ↔ ∀x ∈ A ∀yφ)
4		ralv 2872	. 2 ⊢ (∀y ∈ V ∀x ∈ A φ ↔ ∀y∀x ∈ A φ)
5	1, 3, 4	3bitr3i 266	1 ⊢ (∀x ∈ A ∀yφ ↔ ∀y∀x ∈ A φ)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 176 ∀wal 1540 ∀wral 2614 Vcvv 2859

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 2478 df-ral 2619 df-v 2861

This theorem is referenced by: uniiunlem 3353 iunss 4007 nnadjoinpw 4521 funimass4 5368 clos1induct 5880 dfnnc3 5885

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