Theorem sbsbc 3740

Description: Show that df-sb 2068 and df-sbc 3737 are equivalent when the class term 𝐴 in df-sbc 3737 is a setvar variable. This theorem lets us reuse theorems based on df-sb 2068 for proofs involving df-sbc 3737. (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 2731	. 2 ⊢ 𝑦 = 𝑦
2		dfsbcq2 3739	. 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 2067  [wsbc 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3737

This theorem is referenced by:  spsbc  3749  sbcid  3753  sbccow  3759  sbcco  3762  sbcco2  3763  sbcie2g  3777  eqsbc1  3783  sbcralt  3818  cbvralcsf  3887  cbvreucsf  3889  cbvrabcsf  3890  sbnfc2  4386  csbab  4387  csbie2df  4390  2nreu  4391  frpoins2fg  6291  tfindes  7793  tfinds2  7794  setinds2f  9640  frins2f  9646  iuninc  32540  suppss2f  32620  fmptdF  32638  disjdsct  32684  esumpfinvalf  34089  measiuns  34230  bnj580  34925  bnj985v  34965  bnj985  34966  xpab  35770  bj-sbeq  36945  bj-sbel1  36949  bj-snsetex  37007  poimirlem25  37695  poimirlem26  37696  fdc1  37796  exlimddvfi  38172  frege52b  43992  frege58c  44024  pm13.194  44515  pm14.12  44524  sbiota1  44537  onfrALTlem1  44651  onfrALTlem1VD  44992  disjinfi  45299  ellimcabssub0  45727  2reu8i  47223  ich2exprop  47581  ichnreuop  47582  ichreuopeq  47583