New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > ralcom4 | GIF version |
Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
ralcom4 | ⊢ (∀x ∈ A ∀yφ ↔ ∀y∀x ∈ A φ) |
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 |
Copyright terms: Public domain | W3C validator |