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Theorem wl-sbal2 31381
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 31379 now. See also sbal2 2229. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (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 31349 . 2  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  x  =  z )
2 wl-sbalnae 31379 . 2  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
31, 2mpdan 666 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   A.wal 1403   [wsb 1763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638  df-sb 1764
This theorem is referenced by: (None)
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