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Theorem 19.36i 2246
Description: Inference associated with 19.36 2245. See 19.36iv 2023 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 2245 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3mpbi 220 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630  wex 1853  wnf 1857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-ex 1854  df-nf 1859
This theorem is referenced by:  spimv1  2262  spim  2399  vtoclf  3398  bj-vtoclf  33214
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