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Theorem ralbi 2439
 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 2349 . 2 xx A (φψ)
2 rsp 2363 . . 3 (x A (φψ) → (x A → (φψ)))
32imp 115 . 2 ((x A (φψ) x A) → (φψ))
41, 3ralbida 2314 1 (x A (φψ) → (x A φx A ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  ∀wral 2300 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305 This theorem is referenced by:  uniiunlem  3022  iineq2  3665  ralrnmpt  5252  f1mpt  5353  mpt22eqb  5552  ralrnmpt2  5557
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