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Theorem 19.31 2140
Description: Theorem 19.31 of [Margaris] p. 90. See 19.31v 1910 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 2139 . 2 (∀𝑥(𝜓𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑))
3 orcom 401 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43albii 1787 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜓𝜑))
5 orcom 401 . 2 ((∀𝑥𝜑𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
62, 4, 53bitr4i 292 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wal 1521  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745  df-nf 1750
This theorem is referenced by:  2eu3  2584
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