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Theorem spimt 2427
Description: Closed theorem form of spim 2428. (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 ax6e 2424 . . . 4 𝑥 𝑥 = 𝑦
2 exim 1918 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑𝜓)))
31, 2mpi 20 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ∃𝑥(𝜑𝜓))
4 19.35 1967 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
53, 4sylib 209 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
6 19.9t 2238 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
76biimpd 220 . 2 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
85, 7sylan9r 500 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635  wex 1859  wnf 1863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-12 2214  ax-13 2420
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-nf 1864
This theorem is referenced by: (None)
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