Theorem sbsbc 2967

Description: Show that df-sb 1763 and df-sbc 2964 are equivalent when the class term A in df-sbc 2964 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1763 for proofs involving df-sbc 2964. (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 2966	. 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 2963

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 2964

This theorem is referenced by:  spsbc  2975  sbcid  2979  sbcco  2985  sbcco2  2986  sbcie2g  2997  eqsbc1  3003  sbcralt  3040  sbcrext  3041  sbnfc2  3118  csbabg  3119  cbvralcsf  3120  cbvrexcsf  3121  cbvreucsf  3122  cbvrabcsf  3123  isarep1  5303  finexdc  6902  ssfirab  6933  zsupcllemstep  11946  bezoutlemmain  11999  bdsbc  14613