Theorem sbcbii 2818

Description: Formula-building inference rule 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 2817	. 2 |- ( T. -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
4	3	trud 1252	1 |- ( [. A / x ]. ph <-> [. A / x ]. ps )

Syntax hints:   <-> wb 98   T. wtru 1244   [.wsbc 2764

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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765

This theorem is referenced by:  eqsbc3r  2819  sbccomlem  2832  sbccom  2833  sbcabel  2839  csbco  2861  sbcnel12g  2867  sbcne12g  2868  sbccsbg  2878  sbccsb2g  2879  csbnestgf  2898  csbabg  2907  sbcssg  3330  sbcrel  4426  difopab  4469  sbcfung  4925  mpt2xopovel  5856