Theorem sbcbii 3046

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 3045	. 2 |- ( T. -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
4	3	mptru 1373	1 |- ( [. A / x ]. ph <-> [. A / x ]. ps )

Syntax hints:   <-> wb 105   T. wtru 1365   [.wsbc 2986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-sbc 2987

This theorem is referenced by:  eqsbc2  3047  sbc3an  3048  sbccomlem  3061  sbccom  3062  sbcabel  3068  csbco  3091  csbcow  3092  sbcnel12g  3098  sbcne12g  3099  sbccsbg  3110  sbccsb2g  3111  csbnestgf  3134  csbabg  3143  sbcssg  3556  sbcrel  4746  difopab  4796  sbcfung  5279  f1od2  6290  mpoxopovel  6296  bezoutlemnewy  12136  bezoutlemstep  12137  bezoutlemmain  12138