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Theorem hblem 2265
Description: Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypothesis
Ref Expression
hblem.1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Assertion
Ref Expression
hblem  |-  ( z  e.  A  ->  A. x  z  e.  A )
Distinct variable groups:    y, A    x, z
Allowed substitution hints:    A( x, z)

Proof of Theorem hblem
StepHypRef Expression
1 hblem.1 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
21hbsb 1929 . 2  |-  ( [ z  /  y ] y  e.  A  ->  A. x [ z  / 
y ] y  e.  A )
3 clelsb3 2262 . 2  |-  ( [ z  /  y ] y  e.  A  <->  z  e.  A )
43albii 1450 . 2  |-  ( A. x [ z  /  y ] y  e.  A  <->  A. x  z  e.  A
)
52, 3, 43imtr3i 199 1  |-  ( z  e.  A  ->  A. x  z  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1333   [wsb 1742    e. wcel 2128
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153
This theorem is referenced by:  nfcrii  2292
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