Theorem elabg 3638

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2406. (Revised by SN, 23-Nov-2022.) Avoid ax-10 2178, ax-11 2194, ax-12 2215. (Revised by SN, 5-Oct-2024.)

Hypothesis

Ref	Expression
elabg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elabg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elabg

Step	Hyp	Ref	Expression
1		elabg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	ax-gen 1818	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3		elabgt 3634	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	2, 3	mpan2 703	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   = wceq 1563   ∈ wcel 2145  {cab 2743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840

This theorem is referenced by:  elab  3641  elab2g  3642  elabd  3643  elab3g  3647  sbcieg  3786  intmin3  4936  elabrexg  7231  finds  7881  elfi  9361  inficl  9373  dffi3  9379  scott0  9848  elgch  10595  nqpr  10987  hashf1lem1  14480  cshword  14816  trclublem  15020  cotrtrclfv  15037  dfiso2  17817  efgcpbllemb  19813  frgpuplem  19830  lspsn  21089  mpfind  22223  pf1ind  22472  eltg  23071  eltg2  23072  islocfin  23631  fbssfi  23951  nosupres  27825  nosupbnd1lem3  27828  nosupbnd1lem5  27830  noinffv  27839  noinfres  27840  noinfbnd1lem3  27843  noinfbnd1lem5  27845  isewlk  29857  elabreximd  32762  abfmpunirn  32905  ellpi  33597  fmlafvel  35743  isfmlasuc  35746  r1peuqusdeg1  36001  poimirlem3  38129  poimirlem25  38151  islshpkrN  39751  sticksstones8  42777  sticksstones9  42778  sticksstones11  42780  sticksstones17  42787  sticksstones18  42788  rhmqusspan  42809  sn-iotalem  42847  setindtrs  43609  frege55lem1c  44499  nzss  44886  afvelrnb  47756  afvelrnb0  47757  dfatco  47849  elsetpreimafvb  47989  isgrim  48503  isgrlim  48603  discsntermlem  50200  basrestermcfolem  50201  setis  50328