Theorem sbcbidv 3087

Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)

Hypothesis

Ref	Expression
sbcbidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
sbcbidv	|- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem sbcbidv

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 |- F/ x ph
2		sbcbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
3	1, 2	sbcbid 3086	1 |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 105   [.wsbc 3028

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  sbcbii  3088  csbcomg  3147  opelopabsb  4348  opelopabgf  4358  opelopabf  4363  sbcfng  5471  sbcfg  5472  uchoice  6283  f1od2  6381  wrd2ind  11255  islmod  14255