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Theorem spimt 1974
 Description: Closed theorem form of spim 1975. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)
Assertion
Ref Expression
spimt ((Ⅎxψ x(x = y → (φψ))) → (xφψ))

Proof of Theorem spimt
StepHypRef Expression
1 nfnf1 1790 . . . . 5 xxψ
2 nfa1 1788 . . . . 5 xxφ
31, 2nfan 1824 . . . 4 x(Ⅎxψ xφ)
4 sp 1747 . . . . . . 7 (xφφ)
54adantl 452 . . . . . 6 ((Ⅎxψ xφ) → φ)
6 nfr 1761 . . . . . . 7 (Ⅎxψ → (ψxψ))
76adantr 451 . . . . . 6 ((Ⅎxψ xφ) → (ψxψ))
85, 7embantd 50 . . . . 5 ((Ⅎxψ xφ) → ((φψ) → xψ))
98imim2d 48 . . . 4 ((Ⅎxψ xφ) → ((x = y → (φψ)) → (x = yxψ)))
103, 9alimd 1764 . . 3 ((Ⅎxψ xφ) → (x(x = y → (φψ)) → x(x = yxψ)))
1110impancom 427 . 2 ((Ⅎxψ x(x = y → (φψ))) → (xφx(x = yxψ)))
12 ax9o 1950 . 2 (x(x = yxψ) → ψ)
1311, 12syl6 29 1 ((Ⅎxψ x(x = y → (φψ))) → (xφψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  spim  1975  equveli  1988
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