Theorem sbsbc 2819

Description: Show that df-sb 1686 and df-sbc 2816 are equivalent when the class term A in df-sbc 2816 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1686 for proofs involving df-sbc 2816. (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 2081	. 2 |- y = y
2		dfsbcq2 2818	. 2 |- ( y = y -> ( [ y / x ] ph <-> [. y / x ]. ph ) )
3	1, 2	ax-mp 7	1 |- ( [ y / x ] ph <-> [. y / x ]. ph )

Syntax hints:   <-> wb 103   [wsb 1685   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-sbc 2816

This theorem is referenced by:  spsbc  2826  sbcid  2830  sbcco  2836  sbcco2  2837  sbcie2g  2847  eqsbc3  2853  sbcralt  2890  sbcrext  2891  sbnfc2  2962  csbabg  2963  cbvralcsf  2964  cbvrexcsf  2965  cbvreucsf  2966  cbvrabcsf  2967  isarep1  5005  zsupcllemstep  10341  bezoutlemmain  10387  bdsbc  10649