Theorem elab3 3498

Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)

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 hint:   𝜑(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 3497	. 2 ⊢ ((𝜓 → 𝐴 ∈ V) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ∈ wcel 2139  {cab 2746  Vcvv 3340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342

This theorem is referenced by:  fvelrnb  6405  elrnmpt2  6938  ovelrn  6975  isfi  8145  isnum2  8961  pm54.43lem  9015  isfin3  9310  isfin5  9313  isfin6  9314  genpelv  10014  iswrd  13493  4sqlem2  15855  vdwapval  15879  isghm  17861  issrng  19052  lspsnel  19205  lspprel  19296  iscss  20229  ellspd  20343  istps  20940  islp  21146  is2ndc  21451  elpt  21577  itg2l  23695  elply  24150  isismt  25628  isline  35528  ispointN  35531  ispsubsp  35534  ispsubclN  35726  islaut  35872  ispautN  35888  istendo  36550  rngunsnply  38245