Theorem sbcbii 3091

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 3090	. 2 |- ( T. -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
4	3	mptru 1406	1 |- ( [. A / x ]. ph <-> [. A / x ]. ps )

Syntax hints:   <-> wb 105   T. wtru 1398   [.wsbc 3031

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3032

This theorem is referenced by:  eqsbc2  3092  sbc3an  3093  sbccomlem  3106  sbccom  3107  sbcabel  3114  csbco  3137  csbcow  3138  sbcnel12g  3144  sbcne12g  3145  sbccsbg  3156  sbccsb2g  3157  csbnestgf  3180  csbabg  3189  sbcssg  3603  sbcrel  4812  difopab  4863  sbcfung  5350  f1od2  6399  mpoxopovel  6406  bezoutlemnewy  12566  bezoutlemstep  12567  bezoutlemmain  12568