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Theorem r19.26m 3049
Description: Version of 19.26 1786 and r19.26 3046 with restricted quantifiers ranging over different classes. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
r19.26m (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))

Proof of Theorem r19.26m
StepHypRef Expression
1 19.26 1786 . 2 (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵𝜓)))
2 df-ral 2901 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2901 . . 3 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
42, 3anbi12i 729 . 2 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓) ↔ (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵𝜓)))
51, 4bitr4i 266 1 (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473  wcel 1977  wral 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728
This theorem depends on definitions:  df-bi 196  df-an 385  df-ral 2901
This theorem is referenced by: (None)
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