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Theorem 19.27 2133
Description: Theorem 19.27 of [Margaris] p. 90. See 19.27v 1964 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
19.27.1 𝑥𝜓
Assertion
Ref Expression
19.27 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Proof of Theorem 19.27
StepHypRef Expression
1 19.26 1838 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.27.1 . . . 4 𝑥𝜓
3219.3 2107 . . 3 (∀𝑥𝜓𝜓)
43anbi2i 730 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
51, 4bitri 264 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  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-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  aaan  2206
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