Theorem elrab3 2960

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 2959	. 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 2801

This theorem is referenced by:  unimax  3922  undifexmid  4278  frind  4444  ordtriexmidlem2  4613  ordtriexmid  4614  ontriexmidim  4615  ordtri2orexmid  4616  onsucelsucexmid  4623  0elsucexmid  4658  ordpwsucexmid  4663  ordtri2or2exmid  4664  ontri2orexmidim  4665  canth  5961  acexmidlema  6001  acexmidlemb  6002  isnumi  7370  genpelvl  7715  genpelvu  7716  cauappcvgprlemladdru  7859  cauappcvgprlem1  7862  caucvgprlem1  7882  sup3exmid  9120  supinfneg  9807  infsupneg  9808  supminfex  9809  ublbneg  9825  negm  9827  infssuzex  10470  hashinfuni  11016  gcddvds  12505  dvdslegcd  12506  bezoutlemsup  12551  uzwodc  12579  lcmval  12606  dvdslcm  12612  isprm2lem  12659