Theorem elab 2681

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 1418	. 2 ⊢ Ⅎxψ
2		elab.1	. 2 ⊢ A ∈ V
3		elab.2	. 2 ⊢ (x = A → (φ ↔ ψ))
4	1, 2, 3	elabf 2680	1 ⊢ (A ∈ {x ∣ φ} ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553

This theorem is referenced by:  ralab  2695  rexab  2697  intab  3635  dfiin2g  3681  dfiunv2  3684  uniuni  4149  peano5  4264  finds  4266  finds2  4267  funcnvuni  4911  fun11iun  5090  elabrex  5340  abrexco  5341  indpi  6326  nqprm  6525  nqprrnd  6526  nqprdisj  6527  nqprloc  6528  nqprl  6533  cauappcvgprlem2  6632  caucvgprlem2  6651  1nn  7706  peano2nn  7707  dfuzi  8124  bj-ssom  9395