Theorem elrab3 2963

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 2962	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
3	2	baib 926	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397   ∈ wcel 2202  {crab 2514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804

This theorem is referenced by:  unimax  3927  undifexmid  4283  frind  4449  ordtriexmidlem2  4618  ordtriexmid  4619  ontriexmidim  4620  ordtri2orexmid  4621  onsucelsucexmid  4628  0elsucexmid  4663  ordpwsucexmid  4668  ordtri2or2exmid  4669  ontri2orexmidim  4670  canth  5968  acexmidlema  6008  acexmidlemb  6009  isnumi  7385  genpelvl  7731  genpelvu  7732  cauappcvgprlemladdru  7875  cauappcvgprlem1  7878  caucvgprlem1  7898  sup3exmid  9136  supinfneg  9828  infsupneg  9829  supminfex  9830  ublbneg  9846  negm  9848  infssuzex  10492  hashinfuni  11038  gcddvds  12533  dvdslegcd  12534  bezoutlemsup  12579  uzwodc  12607  lcmval  12634  dvdslcm  12640  isprm2lem  12687