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Theorem tz6.12c 5901
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 2293 . . . 4  |-  ( E! y  A F y  ->  E. y  A F y )
2 nfeu1 2279 . . . . . 6  |-  F/ y E! y  A F y
3 nfv 1755 . . . . . 6  |-  F/ y  A F ( F `
 A )
42, 3nfim 1980 . . . . 5  |-  F/ y ( E! y  A F y  ->  A F ( F `  A ) )
5 tz6.12-1 5898 . . . . . . . 8  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
65expcom 436 . . . . . . 7  |-  ( E! y  A F y  ->  ( A F y  ->  ( F `  A )  =  y ) )
7 breq2 4427 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  ( A F ( F `  A )  <->  A F
y ) )
87biimprd 226 . . . . . . 7  |-  ( ( F `  A )  =  y  ->  ( A F y  ->  A F ( F `  A ) ) )
96, 8syli 38 . . . . . 6  |-  ( E! y  A F y  ->  ( A F y  ->  A F
( F `  A
) ) )
109com12 32 . . . . 5  |-  ( A F y  ->  ( E! y  A F
y  ->  A F
( F `  A
) ) )
114, 10exlimi 1972 . . . 4  |-  ( E. y  A F y  ->  ( E! y  A F y  ->  A F ( F `  A ) ) )
121, 11mpcom 37 . . 3  |-  ( E! y  A F y  ->  A F ( F `  A ) )
1312, 7syl5ibcom 223 . 2  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  ->  A F
y ) )
1413, 6impbid 193 1  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  <->  A F y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   E.wex 1657   E!weu 2269   class class class wbr 4423   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-iota 5565  df-fv 5609
This theorem is referenced by:  tz6.12i  5902  fnbrfvb  5922
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