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Theorem 19.36i 2223
Description: Inference associated with 19.36 2222. See 19.36iv 1938 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
Hypotheses
Ref Expression
19.36.1 𝑥𝜓
19.36i.2 𝑥(𝜑𝜓)
Assertion
Ref Expression
19.36i (∀𝑥𝜑𝜓)

Proof of Theorem 19.36i
StepHypRef Expression
1 19.36i.2 . 2 𝑥(𝜑𝜓)
2 19.36.1 . . 3 𝑥𝜓
3219.36 2222 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3mpbi 231 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  wex 1771  wnf 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-ex 1772  df-nf 1776
This theorem is referenced by:  spimfv  2231  spim  2396  vtoclf  3556  bj-vtoclf  34128
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