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Theorem spimt 1606
 Description: Closed theorem form of spim 1608. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
Assertion
Ref Expression
spimt ((Ⅎxψ x(x = y → (φψ))) → (xφψ))

Proof of Theorem spimt
StepHypRef Expression
1 a9e 1568 . . . 4 x x = y
2 exim 1472 . . . 4 (x(x = y → (φψ)) → (x x = yx(φψ)))
31, 2mpi 15 . . 3 (x(x = y → (φψ)) → x(φψ))
4 19.35-1 1497 . . 3 (x(φψ) → (xφxψ))
53, 4syl 14 . 2 (x(x = y → (φψ)) → (xφxψ))
6 19.9t 1515 . . 3 (Ⅎxψ → (xψψ))
76biimpd 132 . 2 (Ⅎxψ → (xψψ))
85, 7sylan9r 392 1 ((Ⅎxψ x(x = y → (φψ))) → (xφψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1226   = wceq 1228  Ⅎwnf 1329  ∃wex 1362 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-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-i9 1404  ax-ial 1409 This theorem depends on definitions:  df-bi 110  df-nf 1330 This theorem is referenced by:  spimd  7012
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