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  3921  undifexmid  4276  frind  4442  ordtriexmidlem2  4611  ordtriexmid  4612  ontriexmidim  4613  ordtri2orexmid  4614  onsucelsucexmid  4621  0elsucexmid  4656  ordpwsucexmid  4661  ordtri2or2exmid  4662  ontri2orexmidim  4663  canth  5951  acexmidlema  5991  acexmidlemb  5992  isnumi  7350  genpelvl  7695  genpelvu  7696  cauappcvgprlemladdru  7839  cauappcvgprlem1  7842  caucvgprlem1  7862  sup3exmid  9100  supinfneg  9786  infsupneg  9787  supminfex  9788  ublbneg  9804  negm  9806  infssuzex  10448  hashinfuni  10994  gcddvds  12479  dvdslegcd  12480  bezoutlemsup  12525  uzwodc  12553  lcmval  12580  dvdslcm  12586  isprm2lem  12633