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Theorem r19.26m 3065
Description: Version of 19.26 1797 and r19.26 3062 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 1797 . 2 (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵𝜓)))
2 df-ral 2916 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2916 . . 3 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
42, 3anbi12i 733 . 2 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓) ↔ (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵𝜓)))
51, 4bitr4i 267 1 (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480  wcel 1989  wral 2911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736
This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2916
This theorem is referenced by: (None)
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