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Theorem spcimegft 2608
Description: A closed version of spcimegf 2611. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1 xψ
spcimgft.2 xA
Assertion
Ref Expression
spcimegft (x(x = A → (ψφ)) → (A B → (ψxφ)))

Proof of Theorem spcimegft
StepHypRef Expression
1 elex 2543 . 2 (A BA V)
2 spcimgft.2 . . . . 5 xA
32issetf 2540 . . . 4 (A V ↔ x x = A)
4 exim 1472 . . . 4 (x(x = A → (ψφ)) → (x x = Ax(ψφ)))
53, 4syl5bi 141 . . 3 (x(x = A → (ψφ)) → (A V → x(ψφ)))
6 spcimgft.1 . . . 4 xψ
7619.37-1 1546 . . 3 (x(ψφ) → (ψxφ))
85, 7syl6 29 . 2 (x(x = A → (ψφ)) → (A V → (ψxφ)))
91, 8syl5 28 1 (x(x = A → (ψφ)) → (A B → (ψxφ)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226   = wceq 1228  wnf 1329  wex 1362   wcel 1374  wnfc 2147  Vcvv 2535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537
This theorem is referenced by:  spcegft  2609  spcimegf  2611  spcimedv  2616
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