Theorem sbcbii 3065

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 3064	. 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 3005

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-sbc 3006

This theorem is referenced by:  eqsbc2  3066  sbc3an  3067  sbccomlem  3080  sbccom  3081  sbcabel  3088  csbco  3111  csbcow  3112  sbcnel12g  3118  sbcne12g  3119  sbccsbg  3130  sbccsb2g  3131  csbnestgf  3154  csbabg  3163  sbcssg  3577  sbcrel  4779  difopab  4829  sbcfung  5314  f1od2  6344  mpoxopovel  6350  bezoutlemnewy  12432  bezoutlemstep  12433  bezoutlemmain  12434