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Theorem tz6.12c 5878
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 2298 . . . 4  |-  ( E! y  A F y  ->  E. y  A F y )
2 nfeu1 2283 . . . . . 6  |-  F/ y E! y  A F y
3 nfv 1678 . . . . . 6  |-  F/ y  A F ( F `
 A )
42, 3nfim 1862 . . . . 5  |-  F/ y ( E! y  A F y  ->  A F ( F `  A ) )
5 tz6.12-1 5875 . . . . . . . 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 4446 . . . . . . . 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 1854 . . . 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 1374   E.wex 1591   E!weu 2270   class class class wbr 4442   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589
This theorem is referenced by:  tz6.12i  5879  fnbrfvb  5901
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