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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 yφyx A φ)
Distinct variable groups:   x,y   y,A
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem ralcom4
StepHypRef Expression
1 ralcom 2771 . 2 (x A y V φy V x A φ)
2 ralv 2872 . . 3 (y V φyφ)
32ralbii 2638 . 2 (x A y V φx A yφ)
4 ralv 2872 . 2 (y V x A φyx A φ)
51, 3, 43bitr3i 266 1 (x A yφyx 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-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 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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