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Theorem bj-spimtv 34019
Description: Version of spimt 2399 with a disjoint variable condition, which does not require ax-13 2385. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-spimtv ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-spimtv
StepHypRef Expression
1 ax6ev 1965 . . . 4 𝑥 𝑥 = 𝑦
2 exim 1827 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑𝜓)))
31, 2mpi 20 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ∃𝑥(𝜑𝜓))
4 19.35 1871 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
53, 4sylib 219 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
6 19.9t 2197 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
76biimpd 230 . 2 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
85, 7sylan9r 509 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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