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Theorem sbcth2 3389
Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
sbcth2.1  |-  ( x  e.  B  ->  ph )
Assertion
Ref Expression
sbcth2  |-  ( A  e.  B  ->  [. A  /  x ]. ph )
Distinct variable group:    x, B
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem sbcth2
StepHypRef Expression
1 sbcth2.1 . . 3  |-  ( x  e.  B  ->  ph )
21rgen 2792 . 2  |-  A. x  e.  B  ph
3 rspsbc 3384 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
42, 3mpi 21 1  |-  ( A  e.  B  ->  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   A.wral 2782   [.wsbc 3305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-v 3089  df-sbc 3306
This theorem is referenced by: (None)
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