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Theorem rsp2 2482
Description: Restricted specialization. (Contributed by NM, 11-Feb-1997.)
Assertion
Ref Expression
rsp2 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))

Proof of Theorem rsp2
StepHypRef Expression
1 rsp 2480 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 rsp 2480 . . 3 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
31, 2syl6 33 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → (𝑦𝐵𝜑)))
43impd 252 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1480  wral 2416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-4 1487
This theorem depends on definitions:  df-bi 116  df-ral 2421
This theorem is referenced by:  ralidm  3463  sowlin  4242  cnmpt21  12469  cnmpt2t  12471  cnmpt22  12472  cnmptcom  12476
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