Theorem axrepndlem1 8988

Description: Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Assertion

Ref	Expression
axrepndlem1

Distinct variable groups:   ,   ,

Proof of Theorem axrepndlem1

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axrep2 4565	. 2
2		nfnae 2058	. . 3
3		nfnae 2058	. . . . 5
4		nfnae 2058	. . . . . 6
5		nfs1v 2181	. . . . . . . 8
6	5	a1i 11	. . . . . . 7
7		nfcvd 2620	. . . . . . . 8
8		nfcvf2 2645	. . . . . . . 8
9	7, 8	nfeqd 2626	. . . . . . 7
10	6, 9	nfimd 1917	. . . . . 6
11		sbequ12r 1993	. . . . . . . 8
12		equequ1 1798	. . . . . . . 8
13	11, 12	imbi12d 320	. . . . . . 7
14	13	a1i 11	. . . . . 6
15	4, 10, 14	cbvald 2025	. . . . 5
16	3, 15	exbid 1886	. . . 4
17		nfvd 1708	. . . . . 6
18	8	nfcrd 2625	. . . . . . . 8
19	3, 6	nfald 1951	. . . . . . . 8
20	18, 19	nfand 1925	. . . . . . 7
21	2, 20	nfexd 1952	. . . . . 6
22	17, 21	nfbid 1933	. . . . 5
23		elequ1 1821	. . . . . . . 8
24	23	adantl 466	. . . . . . 7
25		nfeqf2 2041	. . . . . . . . . . 11
26	3, 25	nfan1 1927	. . . . . . . . . 10
27	11	adantl 466	. . . . . . . . . 10
28	26, 27	albid 1885	. . . . . . . . 9
29	28	anbi2d 703	. . . . . . . 8
30	29	exbidv 1714	. . . . . . 7
31	24, 30	bibi12d 321	. . . . . 6
32	31	ex 434	. . . . 5
33	4, 22, 32	cbvald 2025	. . . 4
34	16, 33	imbi12d 320	. . 3
35	2, 34	exbid 1886	. 2
36	1, 35	mpbii 211	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  F/wnf 1616  [wsb 1739

This theorem is referenced by:  axrepndlem2  8989

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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563

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