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Theorem 19.31r 1612
 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 1611 . 2 ((𝜓 ∨ ∀𝑥𝜑) → ∀𝑥(𝜓𝜑))
3 orcom 680 . 2 ((∀𝑥𝜑𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
4 orcom 680 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
54albii 1400 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜓𝜑))
62, 3, 53imtr4i 199 1 ((∀𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 662  ∀wal 1283  Ⅎwnf 1390 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-gen 1379  ax-4 1441 This theorem depends on definitions:  df-bi 115  df-nf 1391 This theorem is referenced by: (None)
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