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Theorem pm10.55 39070
Description: Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.55 ((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))

Proof of Theorem pm10.55
StepHypRef Expression
1 exsimpl 1944 . . 3 (∃𝑥(𝜑𝜓) → ∃𝑥𝜑)
21anim1i 593 . 2 ((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))
3 exintr 1968 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
43imdistanri 729 . 2 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)))
52, 4impbii 199 1 ((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  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-an 385  df-ex 1854
This theorem is referenced by: (None)
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