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Theorem 19.41rg 41674
 Description: Closed form of right-to-left implication of 19.41 2235, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 42026. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
19.41rg (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))

Proof of Theorem 19.41rg
StepHypRef Expression
1 sp 2180 . . . 4 (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓))
2 pm3.21 475 . . . . . . 7 (𝜓 → (𝜑 → (𝜑𝜓)))
32a1i 11 . . . . . 6 ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
43al2imi 1817 . . . . 5 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
5 exim 1835 . . . . 5 (∀𝑥(𝜑 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
64, 5syl6 35 . . . 4 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))))
71, 6syld 47 . . 3 (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))))
87com23 86 . 2 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))))
98impd 414 1 (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  ax6e2nd  41682  ax6e2ndVD  42032  ax6e2ndALT  42054
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