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Theorem elabgf 3380
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 2795 . . . 4 𝑥{𝑥𝜑}
31, 2nfel 2806 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
4 elabgf.2 . . 3 𝑥𝜓
53, 4nfbi 1873 . 2 𝑥(𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
6 eleq1 2718 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
7 elabgf.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7bibi12d 334 . 2 (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)))
9 abid 2639 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
101, 5, 8, 9vtoclgf 3295 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wnf 1748  wcel 2030  {cab 2637  wnfc 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233
This theorem is referenced by:  elabf  3381  elabg  3383  elab3gf  3388  elrabf  3392
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