Theorem sbcbii 2963

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 2962	. 2 |- ( T. -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )
4	3	mptru 1340	1 |- ( [. A / x ]. ph <-> [. A / x ]. ps )

Syntax hints:   <-> wb 104   T. wtru 1332   [.wsbc 2904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-sbc 2905

This theorem is referenced by:  eqsbc3r  2964  sbc3an  2965  sbccomlem  2978  sbccom  2979  sbcabel  2985  csbco  3008  sbcnel12g  3014  sbcne12g  3015  sbccsbg  3026  sbccsb2g  3027  csbnestgf  3047  csbabg  3056  sbcssg  3467  sbcrel  4620  difopab  4667  sbcfung  5142  f1od2  6125  mpoxopovel  6131  bezoutlemnewy  11673  bezoutlemstep  11674  bezoutlemmain  11675