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Theorem tz6.12c 5524
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 2049 . . . 4  |-  ( E! y  A F y  ->  E. y  A F y )
2 nfeu1 2030 . . . . . 6  |-  F/ y E! y  A F y
3 nfv 1521 . . . . . 6  |-  F/ y  A F ( F `
 A )
42, 3nfim 1565 . . . . 5  |-  F/ y ( E! y  A F y  ->  A F ( F `  A ) )
5 tz6.12-1 5521 . . . . . . . 8  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
65expcom 115 . . . . . . 7  |-  ( E! y  A F y  ->  ( A F y  ->  ( F `  A )  =  y ) )
7 breq2 3991 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  ( A F ( F `  A )  <->  A F
y ) )
87biimprd 157 . . . . . . 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 30 . . . . 5  |-  ( A F y  ->  ( E! y  A F
y  ->  A F
( F `  A
) ) )
114, 10exlimi 1587 . . . 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 154 . 2  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  ->  A F
y ) )
1413, 6impbid 128 1  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  <->  A F y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348   E.wex 1485   E!weu 2019   class class class wbr 3987   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204
This theorem is referenced by:  fnbrfvb  5535
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