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Theorem rspcsbela 2899
Description: Special case related to rspsbc 2834. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
Assertion
Ref Expression
rspcsbela ((A B x B 𝐶 𝐷) → A / x𝐶 𝐷)
Distinct variable groups:   x,B   x,𝐷
Allowed substitution hints:   A(x)   𝐶(x)

Proof of Theorem rspcsbela
StepHypRef Expression
1 rspsbc 2834 . . 3 (A B → (x B 𝐶 𝐷[A / x]𝐶 𝐷))
2 sbcel1g 2863 . . 3 (A B → ([A / x]𝐶 𝐷A / x𝐶 𝐷))
31, 2sylibd 138 . 2 (A B → (x B 𝐶 𝐷A / x𝐶 𝐷))
43imp 115 1 ((A B x B 𝐶 𝐷) → A / x𝐶 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  wral 2300  [wsbc 2758  csb 2846
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by: (None)
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