Theorem sbsbc 2966

Description: Show that df-sb 1763 and df-sbc 2963 are equivalent when the class term A in df-sbc 2963 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1763 for proofs involving df-sbc 2963. (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 2177	. 2 |- y = y
2		dfsbcq2 2965	. 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 1762   [.wsbc 2962

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2963

This theorem is referenced by:  spsbc  2974  sbcid  2978  sbcco  2984  sbcco2  2985  sbcie2g  2996  eqsbc1  3002  sbcralt  3039  sbcrext  3040  sbnfc2  3117  csbabg  3118  cbvralcsf  3119  cbvrexcsf  3120  cbvreucsf  3121  cbvrabcsf  3122  isarep1  5299  finexdc  6897  ssfirab  6928  zsupcllemstep  11936  bezoutlemmain  11989  bdsbc  14381