Theorem elab3 3663

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 3662	. 2 ⊢ ((𝜓 → 𝐴 ∈ 𝑉) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107  {cab 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808

This theorem is referenced by:  fvelrnb  6935  elrnmpo  7537  ovelrn  7577  isfi  8984  isnum2  9951  pm54.43lem  10006  isfin3  10302  isfin5  10305  isfin6  10306  genpelv  11006  iswrd  14521  4sqlem2  16954  vdwapval  16978  isghm  19183  isghmOLD  19184  issrng  20789  ellspsn  20945  lspprel  21037  iscss  21628  ellspd  21747  istps  22857  islp  23063  is2ndc  23369  elpt  23495  itg2l  25667  elply  26137  isismt  28445  bj-ififc  36521  isline  39679  ispointN  39682  ispsubsp  39685  ispsubclN  39877  islaut  40023  ispautN  40039  istendo  40700  sn-isghm  42621  rngunsnply  43118