Theorem opelopabgf 4772

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4770 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.)

Hypotheses

Ref	Expression
opelopabgf.x
opelopabgf.y
opelopabgf.1
opelopabgf.2

Assertion

Ref	Expression
opelopabgf

Distinct variable groups:   ,,   ,,

Proof of Theorem opelopabgf

Step	Hyp	Ref	Expression
1		opelopabsb 4762	. . 3
2	1	a1i 11	. 2
3		nfcv 2619	. . . . 5
4		opelopabgf.x	. . . . 5
5	3, 4	nfsbc 3349	. . . 4
6		opelopabgf.1	. . . . 5
7	6	sbcbidv 3386	. . . 4
8	5, 7	sbciegf 3359	. . 3
9	8	adantr 465	. 2
10		opelopabgf.y	. . . 4
11		opelopabgf.2	. . . 4
12	10, 11	sbciegf 3359	. . 3
13	12	adantl 466	. 2
14	2, 9, 13	3bitrd 279	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  F/wnf 1616  e.wcel 1818  [.wsbc 3327  <.cop 4035  {copab 4509

This theorem is referenced by:  oprabv  6345

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511