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Theorem 19.41rg 39291
Description: Closed form of right-to-left implication of 19.41 2259, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 39660. (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 2207 . . . 4 (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓))
2 pm3.21 457 . . . . . . 7 (𝜓 → (𝜑 → (𝜑𝜓)))
32a1i 11 . . . . . 6 ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
43al2imi 1891 . . . . 5 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
5 exim 1909 . . . . 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 396 1 (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by:  ax6e2nd  39299  ax6e2ndVD  39666  ax6e2ndALT  39688
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