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Theorem rsp2 3276
Description: Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.)
Assertion
Ref Expression
rsp2 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))

Proof of Theorem rsp2
StepHypRef Expression
1 rsp 3246 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 rsp 3246 . . 3 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
31, 2syl6 35 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → (𝑦𝐵𝜑)))
43impd 410 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-ral 3061
This theorem is referenced by:  ralcom2  3376  disjxiun  5139  mpocurryd  8295  cmncom  19817  cnmpt21  23680  cnmpt2t  23682  cnmpt22  23683  cnmptcom  23687  frgrwopreglem5ALT  30342  htthlem  30937  qsidomlem2  33482  cplgredgex  35127  disjlem14  38800  prtlem14  38876  islptre  45639  sprsymrelfolem2  47485
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