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Theorem tz6.12f 5708
Description: Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
Hypothesis
Ref Expression
tz6.12f.1  |-  F/_ y F
Assertion
Ref Expression
tz6.12f  |-  ( (
<. A ,  y >.  e.  F  /\  E! y
<. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
Distinct variable group:    y, A
Allowed substitution hint:    F( y)

Proof of Theorem tz6.12f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 opeq2 4060 . . . . 5  |-  ( z  =  y  ->  <. A , 
z >.  =  <. A , 
y >. )
21eleq1d 2509 . . . 4  |-  ( z  =  y  ->  ( <. A ,  z >.  e.  F  <->  <. A ,  y
>.  e.  F ) )
3 tz6.12f.1 . . . . . . 7  |-  F/_ y F
43nfel2 2591 . . . . . 6  |-  F/ y
<. A ,  z >.  e.  F
5 nfv 1673 . . . . . 6  |-  F/ z
<. A ,  y >.  e.  F
64, 5, 2cbveu 2296 . . . . 5  |-  ( E! z <. A ,  z
>.  e.  F  <->  E! y <. A ,  y >.  e.  F )
76a1i 11 . . . 4  |-  ( z  =  y  ->  ( E! z <. A ,  z
>.  e.  F  <->  E! y <. A ,  y >.  e.  F ) )
82, 7anbi12d 710 . . 3  |-  ( z  =  y  ->  (
( <. A ,  z
>.  e.  F  /\  E! z <. A ,  z
>.  e.  F )  <->  ( <. A ,  y >.  e.  F  /\  E! y <. A , 
y >.  e.  F ) ) )
9 eqeq2 2452 . . 3  |-  ( z  =  y  ->  (
( F `  A
)  =  z  <->  ( F `  A )  =  y ) )
108, 9imbi12d 320 . 2  |-  ( z  =  y  ->  (
( ( <. A , 
z >.  e.  F  /\  E! z <. A ,  z
>.  e.  F )  -> 
( F `  A
)  =  z )  <-> 
( ( <. A , 
y >.  e.  F  /\  E! y <. A ,  y
>.  e.  F )  -> 
( F `  A
)  =  y ) ) )
11 tz6.12 5707 . 2  |-  ( (
<. A ,  z >.  e.  F  /\  E! z
<. A ,  z >.  e.  F )  ->  ( F `  A )  =  z )
1210, 11chvarv 1958 1  |-  ( (
<. A ,  y >.  e.  F  /\  E! y
<. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E!weu 2253   F/_wnfc 2566   <.cop 3883   ` cfv 5418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426
This theorem is referenced by: (None)
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