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Theorem nfimad 4885
Description: Deduction version of bound-variable hypothesis builder nfima 4884. (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 2285 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2285 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfima 4884 . 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 2282 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2282 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1544 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 2847 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109imaeq1d 4875 . . . . 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 2847 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211imaeq2d 4876 . . . . 5  |-  ( F/_ x B  ->  ( A
" { z  | 
A. x  z  e.  B } )  =  ( A " B
) )
1310, 12sylan9eq 2190 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  -> 
( { z  | 
A. x  z  e.  A } " {
z  |  A. x  z  e.  B }
)  =  ( A
" B ) )
148, 13nfceqdf 2278 . . 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 408 . 2  |-  ( ph  ->  ( F/_ x ( { z  |  A. x  z  e.  A } " { z  | 
A. x  z  e.  B } )  <->  F/_ x ( A " B ) ) )
163, 15mpbii 147 1  |-  ( ph  -> 
F/_ x ( A
" B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329    e. wcel 1480   {cab 2123   F/_wnfc 2266   "cima 4537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547
This theorem is referenced by: (None)
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