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Theorem tz6.12c 5808
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 2288 . . . 4  |-  ( E! y  A F y  ->  E. y  A F y )
2 nfeu1 2273 . . . . . 6  |-  F/ y E! y  A F y
3 nfv 1674 . . . . . 6  |-  F/ y  A F ( F `
 A )
42, 3nfim 1855 . . . . 5  |-  F/ y ( E! y  A F y  ->  A F ( F `  A ) )
5 tz6.12-1 5805 . . . . . . . 8  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
65expcom 435 . . . . . . 7  |-  ( E! y  A F y  ->  ( A F y  ->  ( F `  A )  =  y ) )
7 breq2 4394 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  ( A F ( F `  A )  <->  A F
y ) )
87biimprd 223 . . . . . . 7  |-  ( ( F `  A )  =  y  ->  ( A F y  ->  A F ( F `  A ) ) )
96, 8syli 37 . . . . . 6  |-  ( E! y  A F y  ->  ( A F y  ->  A F
( F `  A
) ) )
109com12 31 . . . . 5  |-  ( A F y  ->  ( E! y  A F
y  ->  A F
( F `  A
) ) )
114, 10exlimi 1847 . . . 4  |-  ( E. y  A F y  ->  ( E! y  A F y  ->  A F ( F `  A ) ) )
121, 11mpcom 36 . . 3  |-  ( E! y  A F y  ->  A F ( F `  A ) )
1312, 7syl5ibcom 220 . 2  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  ->  A F
y ) )
1413, 6impbid 191 1  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  <->  A F y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   E.wex 1587   E!weu 2260   class class class wbr 4390   ` cfv 5516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-iota 5479  df-fv 5524
This theorem is referenced by:  tz6.12i  5809  fnbrfvb  5831
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