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Theorem nfimad 5154
Description: Deduction version of bound-variable hypothesis builder nfima 5153. (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 2530 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2530 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfima 5153 . 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 2528 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2528 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1836 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 3048 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109imaeq1d 5144 . . . . 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 3048 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211imaeq2d 5145 . . . . 5  |-  ( F/_ x B  ->  ( A
" { z  | 
A. x  z  e.  B } )  =  ( A " B
) )
1310, 12sylan9eq 2441 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  -> 
( { z  | 
A. x  z  e.  A } " {
z  |  A. x  z  e.  B }
)  =  ( A
" B ) )
148, 13nfceqdf 2524 . . 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 643 . 2  |-  ( ph  ->  ( F/_ x ( { z  |  A. x  z  e.  A } " { z  | 
A. x  z  e.  B } )  <->  F/_ x ( A " B ) ) )
163, 15mpbii 203 1  |-  ( ph  -> 
F/_ x ( A
" B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    e. wcel 1717   {cab 2375   F/_wnfc 2512   "cima 4823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-cnv 4828  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833
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