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Theorem r19.26m 2461
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 1386 . 2 (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵𝜓)))
2 df-ral 2328 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2328 . . 3 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
42, 3anbi12i 441 . 2 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓) ↔ (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵𝜓)))
51, 4bitr4i 180 1 (∀𝑥((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114  df-ral 2328
This theorem is referenced by: (None)
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