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Theorem hblem 2389
Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2414. (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 2051 . 2  |-  ( [ z  /  y ] y  e.  A  ->  A. x [ z  / 
y ] y  e.  A )
3 clelsb3 2387 . 2  |-  ( [ z  /  y ] y  e.  A  <->  z  e.  A )
43albii 1555 . 2  |-  ( A. x [ z  /  y ] y  e.  A  <->  A. x  z  e.  A
)
52, 3, 43imtr3i 256 1  |-  ( z  e.  A  ->  A. x  z  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1529   [wsb 1631    e. wcel 1686
This theorem is referenced by:  nfcrii  2414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-cleq 2278  df-clel 2281
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