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Theorem spimt 1666
Description: Closed theorem form of spim 1668. (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 1627 . . . 4 𝑥 𝑥 = 𝑦
2 exim 1531 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑𝜓)))
31, 2mpi 15 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ∃𝑥(𝜑𝜓))
4 19.35-1 1556 . . 3 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
53, 4syl 14 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
6 19.9t 1574 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
76biimpd 142 . 2 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
85, 7sylan9r 402 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  wnf 1390  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  spimd  10820
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