Theorem elrab3 2961

Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)

Hypothesis

Ref	Expression
elrab.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elrab3	⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrab3

Step	Hyp	Ref	Expression
1		elrab.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	elrab 2960	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
3	2	baib 924	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {crab 2512

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2802

This theorem is referenced by:  unimax  3925  undifexmid  4281  frind  4447  ordtriexmidlem2  4616  ordtriexmid  4617  ontriexmidim  4618  ordtri2orexmid  4619  onsucelsucexmid  4626  0elsucexmid  4661  ordpwsucexmid  4666  ordtri2or2exmid  4667  ontri2orexmidim  4668  canth  5964  acexmidlema  6004  acexmidlemb  6005  isnumi  7377  genpelvl  7722  genpelvu  7723  cauappcvgprlemladdru  7866  cauappcvgprlem1  7869  caucvgprlem1  7889  sup3exmid  9127  supinfneg  9819  infsupneg  9820  supminfex  9821  ublbneg  9837  negm  9839  infssuzex  10483  hashinfuni  11029  gcddvds  12524  dvdslegcd  12525  bezoutlemsup  12570  uzwodc  12598  lcmval  12625  dvdslcm  12631  isprm2lem  12678