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Theorem elabgf 3572
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 2928 . . . 4 𝑥{𝑥𝜑}
31, 2nfel 2938 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
4 elabgf.2 . . 3 𝑥𝜓
53, 4nfbi 1866 . 2 𝑥(𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
6 eleq1 2847 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
7 elabgf.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7bibi12d 338 . 2 (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)))
9 abid 2757 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
101, 5, 8, 9vtoclgf 3478 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  wnf 1746  wcel 2048  {cab 2753  wnfc 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411
This theorem is referenced by:  elabf  3573  elabgOLD  3575  elab3gf  3581  elrabf  3585
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