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Theorem spcimgft 2623
Description: A closed version of spcimgf 2627. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1 xψ
spcimgft.2 xA
Assertion
Ref Expression
spcimgft (x(x = A → (φψ)) → (A B → (xφψ)))

Proof of Theorem spcimgft
StepHypRef Expression
1 elex 2560 . 2 (A BA V)
2 spcimgft.2 . . . . 5 xA
32issetf 2556 . . . 4 (A V ↔ x x = A)
4 exim 1487 . . . 4 (x(x = A → (φψ)) → (x x = Ax(φψ)))
53, 4syl5bi 141 . . 3 (x(x = A → (φψ)) → (A V → x(φψ)))
6 spcimgft.1 . . . 4 xψ
7619.36-1 1560 . . 3 (x(φψ) → (xφψ))
85, 7syl6 29 . 2 (x(x = A → (φψ)) → (A V → (xφψ)))
91, 8syl5 28 1 (x(x = A → (φψ)) → (A B → (xφψ)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   = wceq 1242  wnf 1346  wex 1378   wcel 1390  wnfc 2162  Vcvv 2551
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  spcgft  2624  spcimgf  2627  spcimdv  2631
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