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Theorem r19.26m 2567
 Description: Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (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 1458 . 2 (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵𝜓)))
2 df-ral 2422 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2422 . . 3 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
42, 3anbi12i 456 . 2 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓) ↔ (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵𝜓)))
51, 4bitr4i 186 1 (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330   ∈ wcel 1481  ∀wral 2417 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426 This theorem depends on definitions:  df-bi 116  df-ral 2422 This theorem is referenced by: (None)
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