Theorem sbcbii 3022

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 3021	. 2 |- ( T. -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
4	3	mptru 1362	1 |- ( [. A / x ]. ph <-> [. A / x ]. ps )

Syntax hints:   <-> wb 105   T. wtru 1354   [.wsbc 2962

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2963

This theorem is referenced by:  eqsbc2  3023  sbc3an  3024  sbccomlem  3037  sbccom  3038  sbcabel  3044  csbco  3067  csbcow  3068  sbcnel12g  3074  sbcne12g  3075  sbccsbg  3086  sbccsb2g  3087  csbnestgf  3109  csbabg  3118  sbcssg  3532  sbcrel  4710  difopab  4757  sbcfung  5237  f1od2  6231  mpoxopovel  6237  bezoutlemnewy  11987  bezoutlemstep  11988  bezoutlemmain  11989