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Theorem spcimgft 2930
 Description: A closed version of spcimgf 2932. (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 2867 . 2 (A BA V)
2 spcimgft.2 . . . . 5 xA
32issetf 2864 . . . 4 (A V ↔ x x = A)
4 exim 1575 . . . 4 (x(x = A → (φψ)) → (x x = Ax(φψ)))
53, 4syl5bi 208 . . 3 (x(x = A → (φψ)) → (A V → x(φψ)))
6 spcimgft.1 . . . 4 xψ
7619.36 1871 . . 3 (x(φψ) ↔ (xφψ))
85, 7syl6ib 217 . 2 (x(x = A → (φψ)) → (A V → (xφψ)))
91, 8syl5 28 1 (x(x = A → (φψ)) → (A B → (xφψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by:  spcgft  2931  spcimgf  2932  spcimdv  2936
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