Theorem sbsbc 3733

Description: Show that df-sb 2069 and df-sbc 3730 are equivalent when the class term 𝐴 in df-sbc 3730 is a setvar variable. This theorem lets us reuse theorems based on df-sb 2069 for proofs involving df-sbc 3730. (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 2737	. 2 ⊢ 𝑦 = 𝑦
2		dfsbcq2 3732	. 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 206  [wsb 2068  [wsbc 3729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-clab 2716  df-cleq 2729  df-clel 2812  df-sbc 3730

This theorem is referenced by:  spsbc  3742  sbcid  3746  sbccow  3752  sbcco  3755  sbcco2  3756  sbcie2g  3770  eqsbc1  3776  sbcralt  3811  cbvralcsf  3880  cbvreucsf  3882  cbvrabcsf  3883  sbnfc2  4380  csbab  4381  csbie2df  4384  2nreu  4385  frpoins2fg  6304  tfindes  7809  tfinds2  7810  setinds2f  9666  frins2f  9672  iuninc  32649  suppss2f  32730  fmptdF  32748  disjdsct  32795  esumpfinvalf  34240  measiuns  34381  bnj580  35075  bnj985v  35115  bnj985  35116  xpab  35928  bj-df-sb  36964  bj-sbeq  37228  bj-sbel1  37232  bj-snsetex  37290  poimirlem25  37984  poimirlem26  37985  fdc1  38085  exlimddvfi  38461  frege52b  44338  frege58c  44370  pm13.194  44861  pm14.12  44870  sbiota1  44883  onfrALTlem1  44997  onfrALTlem1VD  45338  disjinfi  45644  ellimcabssub0  46069  2reu8i  47577  ich2exprop  47947  ichnreuop  47948  ichreuopeq  47949