Theorem sbcbidv 3036

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 1539	. 2 |- F/ x ph
2		sbcbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
3	1, 2	sbcbid 3035	1 |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 105   [.wsbc 2977

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-sbc 2978

This theorem is referenced by:  sbcbii  3037  csbcomg  3095  opelopabsb  4278  opelopabf  4292  sbcfng  5382  sbcfg  5383  f1od2  6261  islmod  13624