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 2147  {cab 2741

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839

This theorem is used by:  fvelrnb  6971  elrnmpo  7583  ovelrn  7624  isfi  9033  isnum2  10056  pm54.43lem  10111  isfin3  10404  isfin5  10407  isfin6  10408  genpelv  11115  iswrd  14690  4sqlem2  17159  vdwapval  17183  isghm  19462  issrng  21134  ellspsn  21311  lspprel  21402  iscss  22022  ellspd  22141  istps  23285  islp  23491  is2ndc  23797  elpt  23924  itg2l  26083  elply  26546  isismt  29030  bj-ififc  37643  isline  40999  ispointN  41002  ispsubsp  41005  ispsubclN  41197  islaut  41343  ispautN  41359  istendo  42020  sn-isghm  43888  rngunsnply  44374