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Theorem wl-sbal2 29591
Description: Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 29589 now. (Revised by Wolf Lammen, 25-Jul-2019.) (New usage is discouraged.)
Assertion
Ref Expression
wl-sbal2  |-  ( -. 
A. x  x  =  y  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
Distinct variable group:    x, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem wl-sbal2
StepHypRef Expression
1 wl-naev 29559 . . 3  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  x  =  z )
21ancli 551 . 2  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )
)
3 wl-sbalnae 29589 . 2  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
42, 3syl 16 1  |-  ( -. 
A. x  x  =  y  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377   [wsb 1711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712
This theorem is referenced by: (None)
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