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Theorem bj-spimt2 36767
Description: A step in the proof of spimt 2388. (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-spimt2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ((∃𝑥𝜓𝜓) → (∀𝑥𝜑𝜓)))

Proof of Theorem bj-spimt2
StepHypRef Expression
1 bj-alequex 36766 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ∃𝑥(𝜑𝜓))
2 19.35 1874 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
31, 2sylib 218 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
43imim1d 82 1 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ((∃𝑥𝜓𝜓) → (∀𝑥𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1534  wex 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-12 2174  ax-13 2374
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776
This theorem is referenced by:  bj-cbv3ta  36768
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