Theorem sbsbc 3032

Description: Show that df-sb 1809 and df-sbc 3029 are equivalent when the class term A in df-sbc 3029 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1809 for proofs involving df-sbc 3029. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbsbc	|- ( [ y / x ] ph <-> [. y / x ]. ph )

Proof of Theorem sbsbc

Step	Hyp	Ref	Expression
1		eqid 2229	. 2 |- y = y
2		dfsbcq2 3031	. 2 |- ( y = y -> ( [ y / x ] ph <-> [. y / x ]. ph ) )
3	1, 2	ax-mp 5	1 |- ( [ y / x ] ph <-> [. y / x ]. ph )

Syntax hints:   <-> wb 105   [wsb 1808   [.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-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  spsbc  3040  sbcid  3044  sbcco  3050  sbcco2  3051  sbcie2g  3062  eqsbc1  3068  sbcralt  3105  sbcrext  3106  sbnfc2  3185  csbabg  3186  cbvralcsf  3187  cbvrexcsf  3188  cbvreucsf  3189  cbvrabcsf  3190  isarep1  5407  finexdc  7064  ssfirab  7098  zsupcllemstep  10449  bezoutlemmain  12519  bdsbc  16221