Theorem elabgf 3244

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
elabgf.2
elabgf.3

Assertion

Ref	Expression
elabgf

Proof of Theorem elabgf

Step	Hyp	Ref	Expression
1		elabgf.1	. 2
2		nfab1 2621	. . . 4
3	1, 2	nfel 2632	. . 3
4		elabgf.2	. . 3
5	3, 4	nfbi 1934	. 2
6		eleq1 2529	. . 3
7		elabgf.3	. . 3
8	6, 7	bibi12d 321	. 2
9		abid 2444	. 2
10	1, 5, 8, 9	vtoclgf 3165	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  F/wnf 1616  e.wcel 1818  {cab 2442  F/_wnfc 2605

This theorem is referenced by:  elabf  3245  elabg  3247  elab3gf  3251  elrabf  3255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111