Theorem sbsbc 3046

Description: Show that df-sb 1812 and df-sbc 3043 are equivalent when the class term A in df-sbc 3043 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1812 for proofs involving df-sbc 3043. (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 2232	. 2 |- y = y
2		dfsbcq2 3045	. 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 1811   [.wsbc 3042

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-cleq 2225  df-clel 2228  df-sbc 3043

This theorem is referenced by:  spsbc  3054  sbcid  3058  sbcco  3064  sbcco2  3065  sbcie2g  3076  eqsbc1  3082  sbcralt  3119  sbcrext  3120  sbnfc2  3199  csbabg  3200  cbvralcsf  3201  cbvrexcsf  3202  cbvreucsf  3203  cbvrabcsf  3204  isarep1  5442  finexdc  7160  ssfirab  7197  zsupcllemstep  10589  bezoutlemmain  12694  bdsbc  16628