Theorem elab 2908

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 1542	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2907	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {cab 2182  Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  ralab  2924  rexab  2926  intab  3903  dfiin2g  3949  dfiunv2  3952  uniuni  4486  dcextest  4617  peano5  4634  finds  4636  finds2  4637  funcnvuni  5327  fun11iun  5525  elabrex  5804  abrexco  5806  mapval2  6737  ssenen  6912  snexxph  7016  sbthlem2  7024  indpi  7409  nqprm  7609  nqprrnd  7610  nqprdisj  7611  nqprloc  7612  nqprl  7618  nqpru  7619  cauappcvgprlem2  7727  caucvgprlem2  7747  peano1nnnn  7919  peano2nnnn  7920  1nn  9001  peano2nn  9002  dfuzi  9436  hashfacen  10928  shftfvalg  10983  ovshftex  10984  shftfval  10986  4sqlemafi  12564  lss1d  13939  txdis1cn  14514  bj-ssom  15582