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Theorem elabgf 2850
 Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabgf.1 𝑥𝐴
elabgf.2 𝑥𝜓
elabgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elabgf (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))

Proof of Theorem elabgf
StepHypRef Expression
1 elabgf.1 . 2 𝑥𝐴
2 nfab1 2298 . . . 4 𝑥{𝑥𝜑}
31, 2nfel 2305 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
4 elabgf.2 . . 3 𝑥𝜓
53, 4nfbi 1566 . 2 𝑥(𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
6 eleq1 2217 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
7 elabgf.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7bibi12d 234 . 2 (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)))
9 abid 2142 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
101, 5, 8, 9vtoclgf 2767 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332  Ⅎwnf 1437   ∈ wcel 2125  {cab 2140  Ⅎwnfc 2283 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711 This theorem is referenced by:  elabf  2851  elabg  2854  elab3gf  2858  elrabf  2862  bj-intabssel  13301
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