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Theorem ralbi 2750
 Description: Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)
Assertion
Ref Expression
ralbi (x A (φψ) → (x A φx A ψ))

Proof of Theorem ralbi
StepHypRef Expression
1 nfra1 2664 . 2 xx A (φψ)
2 rsp 2674 . . 3 (x A (φψ) → (x A → (φψ)))
32imp 418 . 2 ((x A (φψ) x A) → (φψ))
41, 3ralbida 2628 1 (x A (φψ) → (x A φx A ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by:  uniiunlem  3353  iineq2  3986
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