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)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812

This theorem is referenced by:  fvelrnb  6902  elrnmpo  7504  ovelrn  7544  isfi  8924  isnum2  9869  pm54.43lem  9924  isfin3  10218  isfin5  10221  isfin6  10222  genpelv  10923  iswrd  14450  4sqlem2  16889  vdwapval  16913  isghm  19156  isghmOLD  19157  issrng  20789  ellspsn  20966  lspprel  21058  iscss  21650  ellspd  21769  istps  22890  islp  23096  is2ndc  23402  elpt  23528  itg2l  25698  elply  26168  isismt  28618  bj-ififc  36803  isline  40109  ispointN  40112  ispsubsp  40115  ispsubclN  40307  islaut  40453  ispautN  40469  istendo  41130  sn-isghm  43025  rngunsnply  43520