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Theorem 2uasban 28328
Description: Distribute the unabbreviated form of proper substitution in and out of a conjunction. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2uasban  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  <->  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
Distinct variable groups:    x, u    y, u    x, v    y,
v
Allowed substitution hints:    ph( x, y, v, u)    ps( x, y, v, u)

Proof of Theorem 2uasban
StepHypRef Expression
1 biid 227 . 2  |-  ( ( E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) )  <->  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
212uasbanh 28327 1  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  <->  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-v 2790
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