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Theorem sbralie 2665
 Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
Hypothesis
Ref Expression
sbralie.1
Assertion
Ref Expression
sbralie
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem sbralie
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cbvralsv 2663 . . 3
21sbbii 1738 . 2
3 nfv 1508 . . 3
4 raleq 2624 . . 3
53, 4sbie 1764 . 2
6 cbvralsv 2663 . . 3
7 nfv 1508 . . . . . 6
87sbco2 1936 . . . . 5
9 nfv 1508 . . . . . 6
10 sbralie.1 . . . . . . . 8
1110bicomd 140 . . . . . . 7
1211equcoms 1684 . . . . . 6
139, 12sbie 1764 . . . . 5
148, 13bitri 183 . . . 4
1514ralbii 2439 . . 3
166, 15bitri 183 . 2
172, 5, 163bitrri 206 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104  wsb 1735  wral 2414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419 This theorem is referenced by: (None)
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