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Theorem 19.31 2235
 Description: Theorem 19.31 of [Margaris] p. 90. See 19.31v 1943 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)
Hypothesis
Ref Expression
19.31.1 𝑥𝜓
Assertion
Ref Expression
19.31 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Proof of Theorem 19.31
StepHypRef Expression
1 19.31.1 . . 3 𝑥𝜓
2119.32 2234 . 2 (∀𝑥(𝜓𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑))
3 orcom 868 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43albii 1822 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜓𝜑))
5 orcom 868 . 2 ((∀𝑥𝜑𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
62, 4, 53bitr4i 307 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 845  ∀wal 1537  Ⅎwnf 1786 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-or 846  df-ex 1783  df-nf 1787 This theorem is referenced by:  2eu3  2675
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