Theorem sbsbc 3048

Description: Show that df-sb 1812 and df-sbc 3045 are equivalent when the class term 𝐴 in df-sbc 3045 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1812 for proofs involving df-sbc 3045. (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 2234	. 2 ⊢ 𝑦 = 𝑦
2		dfsbcq2 3047	. 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 105  [wsb 1811  [wsbc 3044

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 2216

This theorem depends on definitions:  df-bi 117  df-clab 2221  df-cleq 2227  df-clel 2230  df-sbc 3045

This theorem is referenced by:  spsbc  3056  sbcid  3060  sbcco  3066  sbcco2  3067  sbcie2g  3078  eqsbc1  3084  sbcralt  3121  sbcrext  3122  sbnfc2  3201  csbabg  3202  cbvralcsf  3203  cbvrexcsf  3204  cbvreucsf  3205  cbvrabcsf  3206  isarep1  5444  finexdc  7162  ssfirab  7199  zsupcllemstep  10593  bezoutlemmain  12698  bdsbc  16645