Theorem sbcbidv 3013

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 1521	. 2 |- F/ x ph
2		sbcbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
3	1, 2	sbcbid 3012	1 |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 104   [.wsbc 2955

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-sbc 2956

This theorem is referenced by:  sbcbii  3014  csbcomg  3072  opelopabsb  4245  opelopabf  4259  sbcfng  5345  sbcfg  5346  f1od2  6214