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Theorem nfimad 4770
Description: Deduction version of bound-variable hypothesis builder nfima 4769. (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 2234 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2234 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfima 4769 . 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 2231 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2231 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1502 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 2781 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109imaeq1d 4760 . . . . 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 2781 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211imaeq2d 4761 . . . . 5  |-  ( F/_ x B  ->  ( A
" { z  | 
A. x  z  e.  B } )  =  ( A " B
) )
1310, 12sylan9eq 2140 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  -> 
( { z  | 
A. x  z  e.  A } " {
z  |  A. x  z  e.  B }
)  =  ( A
" B ) )
148, 13nfceqdf 2227 . . 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 403 . 2  |-  ( ph  ->  ( F/_ x ( { z  |  A. x  z  e.  A } " { z  | 
A. x  z  e.  B } )  <->  F/_ x ( A " B ) ) )
163, 15mpbii 146 1  |-  ( ph  -> 
F/_ x ( A
" B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    e. wcel 1438   {cab 2074   F/_wnfc 2215   "cima 4431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by: (None)
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