Theorem cbvopab1s 4524

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

Assertion

Ref	Expression
cbvopab1s

Distinct variable groups:   ,,   ,

Proof of Theorem cbvopab1s

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1707	. . . 4
2		nfv 1707	. . . . . 6
3		nfs1v 2181	. . . . . 6
4	2, 3	nfan 1928	. . . . 5
5	4	nfex 1948	. . . 4
6		opeq1 4217	. . . . . . 7
7	6	eqeq2d 2471	. . . . . 6
8		sbequ12 1992	. . . . . 6
9	7, 8	anbi12d 710	. . . . 5
10	9	exbidv 1714	. . . 4
11	1, 5, 10	cbvex 2022	. . 3
12	11	abbii 2591	. 2
13		df-opab 4511	. 2
14		df-opab 4511	. 2
15	12, 13, 14	3eqtr4i 2496	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  [wsb 1739  {cab 2442  <.cop 4035  {copab 4509

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-or 370  df-an 371  df-3an 975  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-rab 2816  df-v 3111  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