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Theorem hblem 2353
Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2378. (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 2070 . 2  |-  ( [ z  /  y ] y  e.  A  ->  A. x [ z  / 
y ] y  e.  A )
3 clelsb3 2351 . 2  |-  ( [ z  /  y ] y  e.  A  <->  z  e.  A )
43albii 1554 . 2  |-  ( A. x [ z  /  y ] y  e.  A  <->  A. x  z  e.  A
)
52, 3, 43imtr3i 258 1  |-  ( z  e.  A  ->  A. x  z  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532    e. wcel 1621   [wsb 1882
This theorem is referenced by:  nfcrii  2378
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-cleq 2246  df-clel 2249
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