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Theorem spcgft 2931
 Description: A closed version of spcgf 2934. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1 xψ
spcimgft.2 xA
Assertion
Ref Expression
spcgft (x(x = A → (φψ)) → (A B → (xφψ)))

Proof of Theorem spcgft
StepHypRef Expression
1 bi1 178 . . . 4 ((φψ) → (φψ))
21imim2i 13 . . 3 ((x = A → (φψ)) → (x = A → (φψ)))
32alimi 1559 . 2 (x(x = A → (φψ)) → x(x = A → (φψ)))
4 spcimgft.1 . . 3 xψ
5 spcimgft.2 . . 3 xA
64, 5spcimgft 2930 . 2 (x(x = A → (φψ)) → (A B → (xφψ)))
73, 6syl 15 1 (x(x = A → (φψ)) → (A B → (xφψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  spcgf  2934  rspct  2948
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