Theorem sbcbii 2898

Description: Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.)

Hypothesis

Ref	Expression
sbcbii.1	|- ( ph <-> ps )

Assertion

Ref	Expression
sbcbii	|- ( [. A / x ]. ph <-> [. A / x ]. ps )

Proof of Theorem sbcbii

Step	Hyp	Ref	Expression
1		sbcbii.1	. . . 4 |- ( ph <-> ps )
2	1	a1i 9	. . 3 |- ( T. -> ( ph <-> ps ) )
3	2	sbcbidv 2897	. 2 |- ( T. -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
4	3	mptru 1298	1 |- ( [. A / x ]. ph <-> [. A / x ]. ps )

Syntax hints:   <-> wb 103   T. wtru 1290   [.wsbc 2840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2841

This theorem is referenced by:  eqsbc3r  2899  sbc3an  2900  sbccomlem  2913  sbccom  2914  sbcabel  2920  csbco  2942  sbcnel12g  2948  sbcne12g  2949  sbccsbg  2959  sbccsb2g  2960  csbnestgf  2980  csbabg  2989  sbcssg  3391  sbcrel  4524  difopab  4569  sbcfung  5039  f1od2  6000  mpt2xopovel  6006  bezoutlemnewy  11263  bezoutlemstep  11264  bezoutlemmain  11265