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Theorem nfimad 5037
Description: Deduction version of bound-variable hypothesis builder nfima 5036. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfimad.2  |-  ( ph  -> 
F/_ x A )
nfimad.3  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfimad  |-  ( ph  -> 
F/_ x ( A
" B ) )

Proof of Theorem nfimad
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2437 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2437 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfima 5036 . 2  |-  F/_ x
( { z  | 
A. x  z  e.  A } " {
z  |  A. x  z  e.  B }
)
4 nfimad.2 . . 3  |-  ( ph  -> 
F/_ x A )
5 nfimad.3 . . 3  |-  ( ph  -> 
F/_ x B )
6 nfnfc1 2435 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2435 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1783 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 2947 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109imaeq1d 5027 . . . . 5  |-  ( F/_ x A  ->  ( { z  |  A. x  z  e.  A } " { z  |  A. x  z  e.  B } )  =  ( A " { z  |  A. x  z  e.  B } ) )
11 abidnf 2947 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211imaeq2d 5028 . . . . 5  |-  ( F/_ x B  ->  ( A
" { z  | 
A. x  z  e.  B } )  =  ( A " B
) )
1310, 12sylan9eq 2348 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  -> 
( { z  | 
A. x  z  e.  A } " {
z  |  A. x  z  e.  B }
)  =  ( A
" B ) )
148, 13nfceqdf 2431 . . 3  |-  ( (
F/_ x A  /\  F/_ x B )  -> 
( F/_ x ( { z  |  A. x  z  e.  A } " { z  |  A. x  z  e.  B } )  <->  F/_ x ( A " B ) ) )
154, 5, 14syl2anc 642 . 2  |-  ( ph  ->  ( F/_ x ( { z  |  A. x  z  e.  A } " { z  | 
A. x  z  e.  B } )  <->  F/_ x ( A " B ) ) )
163, 15mpbii 202 1  |-  ( ph  -> 
F/_ x ( A
" B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    e. wcel 1696   {cab 2282   F/_wnfc 2419   "cima 4708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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