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Theorem bj-spimt2 33040
Description: A step in the proof of spimt 2414. (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 33039 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ∃𝑥(𝜑𝜓))
2 19.35 1956 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
31, 2sylib 208 . 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 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202  ax-13 2407
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852
This theorem is referenced by:  bj-cbv3ta  33041
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