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Theorem 19.31r 1644
Description: One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
Hypothesis
Ref Expression
19.31r.1 𝑥𝜓
Assertion
Ref Expression
19.31r ((∀𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓))

Proof of Theorem 19.31r
StepHypRef Expression
1 19.31r.1 . . 3 𝑥𝜓
2119.32r 1643 . 2 ((𝜓 ∨ ∀𝑥𝜑) → ∀𝑥(𝜓𝜑))
3 orcom 702 . 2 ((∀𝑥𝜑𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
4 orcom 702 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
54albii 1431 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜓𝜑))
62, 3, 53imtr4i 200 1 ((∀𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 682  wal 1314  wnf 1421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-gen 1410  ax-4 1472
This theorem depends on definitions:  df-bi 116  df-nf 1422
This theorem is referenced by: (None)
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