Theorem elab3 3673

Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) (Revised by AV, 16-Aug-2024.)

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 hints:   𝜑(𝑥)   𝑉(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 3672	. 2 ⊢ ((𝜓 → 𝐴 ∈ 𝑉) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cab 2702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802

This theorem is referenced by:  fvelrnb  6956  elrnmpo  7555  ovelrn  7595  isfi  8995  isnum2  9968  pm54.43lem  10023  isfin3  10319  isfin5  10322  isfin6  10323  genpelv  11023  iswrd  14498  4sqlem2  16917  vdwapval  16941  isghm  19174  issrng  20734  lspsnel  20891  lspprel  20983  iscss  21619  ellspd  21740  istps  22866  islp  23074  is2ndc  23380  elpt  23506  itg2l  25689  elply  26159  isismt  28394  bj-ififc  36128  isline  39281  ispointN  39284  ispsubsp  39287  ispsubclN  39479  islaut  39625  ispautN  39641  istendo  40302  rngunsnply  42662