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  3904  dfiin2g  3950  dfiunv2  3953  uniuni  4487  dcextest  4618  peano5  4635  finds  4637  finds2  4638  funcnvuni  5328  fun11iun  5528  elabrex  5807  abrexco  5809  mapval2  6746  ssenen  6921  snexxph  7025  sbthlem2  7033  indpi  7428  nqprm  7628  nqprrnd  7629  nqprdisj  7630  nqprloc  7631  nqprl  7637  nqpru  7638  cauappcvgprlem2  7746  caucvgprlem2  7766  peano1nnnn  7938  peano2nnnn  7939  1nn  9020  peano2nn  9021  dfuzi  9455  hashfacen  10947  shftfvalg  11002  ovshftex  11003  shftfval  11005  4sqlemafi  12591  lss1d  14017  txdis1cn  14600  bj-ssom  15668