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Theorem pm10.57 39090
Description: Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.57 (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ∨ ∃𝑥(𝜑𝜒)))

Proof of Theorem pm10.57
StepHypRef Expression
1 alnex 1855 . . . 4 (∀𝑥 ¬ (𝜑𝜒) ↔ ¬ ∃𝑥(𝜑𝜒))
2 imnan 437 . . . . . 6 ((𝜑 → ¬ 𝜒) ↔ ¬ (𝜑𝜒))
3 pm2.53 387 . . . . . . . 8 ((𝜓𝜒) → (¬ 𝜓𝜒))
43con1d 139 . . . . . . 7 ((𝜓𝜒) → (¬ 𝜒𝜓))
54imim3i 64 . . . . . 6 ((𝜑 → (𝜓𝜒)) → ((𝜑 → ¬ 𝜒) → (𝜑𝜓)))
62, 5syl5bir 233 . . . . 5 ((𝜑 → (𝜓𝜒)) → (¬ (𝜑𝜒) → (𝜑𝜓)))
76al2imi 1892 . . . 4 (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥 ¬ (𝜑𝜒) → ∀𝑥(𝜑𝜓)))
81, 7syl5bir 233 . . 3 (∀𝑥(𝜑 → (𝜓𝜒)) → (¬ ∃𝑥(𝜑𝜒) → ∀𝑥(𝜑𝜓)))
98con1d 139 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → (¬ ∀𝑥(𝜑𝜓) → ∃𝑥(𝜑𝜒)))
109orrd 392 1 (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ∨ ∃𝑥(𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854
This theorem is referenced by: (None)
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