Theorem elab 2664

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 1402	. 2 ⊢ Ⅎxψ
2		elab.1	. 2 ⊢ A ∈ V
3		elab.2	. 2 ⊢ (x = A → (φ ↔ ψ))
4	1, 2, 3	elabf 2663	1 ⊢ (A ∈ {x ∣ φ} ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228   ∈ wcel 1374  {cab 2008  Vcvv 2535

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537

This theorem is referenced by:  ralab  2678  rexab  2680  intab  3618  dfiin2g  3664  dfiunv2  3667  uniuni  4133  peano5  4248  finds  4250  finds2  4251  funcnvuni  4894  fun11iun  5072  elabrex  5322  abrexco  5323  nqprm  6397  nqprrnd  6398  nqprdisj  6399  nqprloc  6400  bj-ssom  7305