Theorem sbcbii 3058

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 3057	. 2 |- ( T. -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
4	3	mptru 1382	1 |- ( [. A / x ]. ph <-> [. A / x ]. ps )

Syntax hints:   <-> wb 105   T. wtru 1374   [.wsbc 2998

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-sbc 2999

This theorem is referenced by:  eqsbc2  3059  sbc3an  3060  sbccomlem  3073  sbccom  3074  sbcabel  3080  csbco  3103  csbcow  3104  sbcnel12g  3110  sbcne12g  3111  sbccsbg  3122  sbccsb2g  3123  csbnestgf  3146  csbabg  3155  sbcssg  3569  sbcrel  4761  difopab  4811  sbcfung  5295  f1od2  6321  mpoxopovel  6327  bezoutlemnewy  12317  bezoutlemstep  12318  bezoutlemmain  12319