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Theorem hblem 2569
Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2595. (Contributed by NM, 21-Jun-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 2280 . 2  |-  ( [ z  /  y ] y  e.  A  ->  A. x [ z  / 
y ] y  e.  A )
3 clelsb3 2567 . 2  |-  ( [ z  /  y ] y  e.  A  <->  z  e.  A )
43albii 1701 . 2  |-  ( A. x [ z  /  y ] y  e.  A  <->  A. x  z  e.  A
)
52, 3, 43imtr3i 273 1  |-  ( z  e.  A  ->  A. x  z  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1452   [wsb 1807    e. wcel 1897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808  df-cleq 2454  df-clel 2457
This theorem is referenced by:  nfcrii  2595  bnj1311  29881
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