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Theorem elabgf 3221
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 2657 . . . 4 𝑥{𝑥𝜑}
31, 2nfel 2667 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
4 elabgf.2 . . 3 𝑥𝜓
53, 4nfbi 2065 . 2 𝑥(𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
6 eleq1 2580 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
7 elabgf.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7bibi12d 333 . 2 (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)))
9 abid 2502 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
101, 5, 8, 9vtoclgf 3141 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wnf 1698  wcel 1938  {cab 2500  wnfc 2642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-v 3079
This theorem is referenced by:  elabf  3222  elabg  3224  elab3gf  3229  elrabf  3233
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