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Theorem sbcth2 3426
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 2824 . 2  |-  A. x  e.  B  ph
3 rspsbc 3421 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
42, 3mpi 17 1  |-  ( A  e.  B  ->  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   A.wral 2814   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-sbc 3332
This theorem is referenced by: (None)
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