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Theorem tz6.12c 5509
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12c  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  <->  A F y ) )
Distinct variable groups:    y, F    y, A

Proof of Theorem tz6.12c
StepHypRef Expression
1 euex 2168 . . . 4  |-  ( E! y  A F y  ->  E. y  A F y )
2 nfeu1 2155 . . . . . 6  |-  F/ y E! y  A F y
3 nfv 1606 . . . . . 6  |-  F/ y  A F ( F `
 A )
42, 3nfim 1771 . . . . 5  |-  F/ y ( E! y  A F y  ->  A F ( F `  A ) )
5 tz6.12-1 5505 . . . . . . . 8  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
65expcom 426 . . . . . . 7  |-  ( E! y  A F y  ->  ( A F y  ->  ( F `  A )  =  y ) )
7 breq2 4029 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  ( A F ( F `  A )  <->  A F
y ) )
87biimprd 216 . . . . . . 7  |-  ( ( F `  A )  =  y  ->  ( A F y  ->  A F ( F `  A ) ) )
96, 8syli 35 . . . . . 6  |-  ( E! y  A F y  ->  ( A F y  ->  A F
( F `  A
) ) )
109com12 29 . . . . 5  |-  ( A F y  ->  ( E! y  A F
y  ->  A F
( F `  A
) ) )
114, 10exlimi 1803 . . . 4  |-  ( E. y  A F y  ->  ( E! y  A F y  ->  A F ( F `  A ) ) )
121, 11mpcom 34 . . 3  |-  ( E! y  A F y  ->  A F ( F `  A ) )
1312, 7syl5ibcom 213 . 2  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  ->  A F
y ) )
1413, 6impbid 185 1  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  <->  A F y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   E.wex 1529    = wceq 1624   E!weu 2145   class class class wbr 4025   ` cfv 5222
This theorem is referenced by:  tz6.12i  5510  fnbrfvb  5525
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fv 5230
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