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Theorem spimth 1601
 Description: Closed theorem form of spim 1604. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)
Assertion
Ref Expression
spimth (x((ψxψ) (x = y → (φψ))) → (xφψ))

Proof of Theorem spimth
StepHypRef Expression
1 imim2 49 . . . . . 6 ((ψxψ) → ((φψ) → (φxψ)))
21imim2d 48 . . . . 5 ((ψxψ) → ((x = y → (φψ)) → (x = y → (φxψ))))
32imp 115 . . . 4 (((ψxψ) (x = y → (φψ))) → (x = y → (φxψ)))
43com23 72 . . 3 (((ψxψ) (x = y → (φψ))) → (φ → (x = yxψ)))
54al2imi 1323 . 2 (x((ψxψ) (x = y → (φψ))) → (xφx(x = yxψ)))
6 ax9o 1566 . 2 (x(x = yxψ) → ψ)
75, 6syl6 29 1 (x((ψxψ) (x = y → (φψ))) → (xφψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1224 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 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-i9 1400  ax-ial 1405 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  equveli  1620
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