Theorem elab 2985

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)

Hypotheses

Ref	Expression
elab.1	⊢ A ∈ V
elab.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
elab	⊢ (A ∈ {x ∣ φ} ↔ ψ)

Distinct variable groups:   ψ,x   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem elab

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxψ
2		elab.1	. 2 ⊢ A ∈ V
3		elab.2	. 2 ⊢ (x = A → (φ ↔ ψ))
4	1, 2, 3	elabf 2984	1 ⊢ (A ∈ {x ∣ φ} ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by:  ralab  2997  rexab  2999  intab  3956  dfiin2g  4000  pwadjoin  4119  dfnnc2  4395  findsd  4410  nnadjoinlem1  4519  tfinnn  4534  vfinncvntsp  4549  vfinspss  4551  vfinspclt  4552  proj1op  4600  proj2op  4601  funcnvuni  5161  fun11iun  5305  tz6.12-2  5346  fnasrn  5417  abrexco  5463  clos1is  5881  mapval2  6018  mucex  6133  ceex  6174  spacis  6288