Theorem elab2 2974

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Hypotheses

Ref	Expression
elab2.1	⊢ 𝐴 ∈ V
elab2.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
elab2.3	⊢ 𝐵 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
elab2	⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elab2

Step	Hyp	Ref	Expression
1		elab2.1	. 2 ⊢ 𝐴 ∈ V
2		elab2.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3		elab2.3	. . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑}
4	2, 3	elab2g 2973	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  Vcvv 2821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is used by:  elpw  3694  elint  3976  opabid  4400  elrn2  5026  elimasn  5157  oprabid  6120  tfrlem3a  6585  tfrcllemsucaccv  6629  tfrcllembxssdm  6631  tfrcllemres  6637  addnqprlemrl  7931  addnqprlemru  7932  addnqprlemfl  7933  addnqprlemfu  7934  mulnqprlemrl  7947  mulnqprlemru  7948  mulnqprlemfl  7949  mulnqprlemfu  7950  ltnqpr  7967  ltnqpri  7968  archpr  8017  cauappcvgprlemladdfu  8028  cauappcvgprlemladdfl  8029  caucvgprlemladdfu  8051  caucvgprprlemopu  8073  suplocexprlemloc  8095  4sqlem12  13211  txuni2  15494