Theorem elab3 2968

Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)

Hypotheses

Ref	Expression
elab3.1	⊢ (𝜓 → 𝐴 ∈ V)
elab3.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elab3	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elab3

Step	Hyp	Ref	Expression
1		elab3.1	. 2 ⊢ (𝜓 → 𝐴 ∈ V)
2		elab3.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	elab3g 2967	. 2 ⊢ ((𝜓 → 𝐴 ∈ V) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {cab 2218  Vcvv 2812

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814

This theorem is referenced by:  elrnmpo  6166  isfi  6999  addnqprlemfl  7870  addnqprlemfu  7871  mulnqprlemfl  7886  mulnqprlemfu  7887  iswrd  11219  4sqlem2  13080  istps  14884  elply  15586