Theorem sbsbc 3051

Description: Show that df-sb 1649 and df-sbc 3048 are equivalent when the class term A in df-sbc 3048 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1649 for proofs involving df-sbc 3048. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbsbc	⊢ ([y / x]φ ↔ [̣y / x]̣φ)

Proof of Theorem sbsbc

Step	Hyp	Ref	Expression
1		eqid 2353	. 2 ⊢ y = y
2		dfsbcq2 3050	. 2 ⊢ (y = y → ([y / x]φ ↔ [̣y / x]̣φ))
3	1, 2	ax-mp 5	1 ⊢ ([y / x]φ ↔ [̣y / x]̣φ)

Syntax hints:   ↔ wb 176  [wsb 1648  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3048

This theorem is referenced by:  spsbc  3059  sbcid  3063  sbcco  3069  sbcco2  3070  sbcie2g  3080  eqsbc1  3086  sbcralt  3119  csbid  3144  sbnfc2  3197  csbabg  3198  cbvralcsf  3199  cbvreucsf  3201  cbvrabcsf  3202