Theorem elab3 3643

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 3642	. 2 ⊢ ((𝜓 → 𝐴 ∈ 𝑉) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2740

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837

This theorem is used by:  fvelrnb  6942  elrnmpo  7553  ovelrn  7594  isfi  8985  isnum2  9954  pm54.43lem  10009  isfin3  10302  isfin5  10305  isfin6  10306  genpelv  11013  iswrd  14584  4sqlem2  17047  vdwapval  17071  isghm  19349  issrng  21016  ellspsn  21193  lspprel  21284  iscss  21902  ellspd  22021  istps  23165  islp  23371  is2ndc  23677  elpt  23804  itg2l  25963  elply  26427  isismt  28884  bj-ififc  37291  isline  40620  ispointN  40623  ispsubsp  40626  ispsubclN  40818  islaut  40964  ispautN  40980  istendo  41641  sn-isghm  43527  rngunsnply  44018