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  4277  frind  4443  ordtriexmidlem2  4612  ordtriexmid  4613  ontriexmidim  4614  ordtri2orexmid  4615  onsucelsucexmid  4622  0elsucexmid  4657  ordpwsucexmid  4662  ordtri2or2exmid  4663  ontri2orexmidim  4664  canth  5958  acexmidlema  5998  acexmidlemb  5999  isnumi  7365  genpelvl  7710  genpelvu  7711  cauappcvgprlemladdru  7854  cauappcvgprlem1  7857  caucvgprlem1  7877  sup3exmid  9115  supinfneg  9802  infsupneg  9803  supminfex  9804  ublbneg  9820  negm  9822  infssuzex  10465  hashinfuni  11011  gcddvds  12499  dvdslegcd  12500  bezoutlemsup  12545  uzwodc  12573  lcmval  12600  dvdslcm  12606  isprm2lem  12653