Theorem cbvixp 7506

Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

Ref	Expression
cbvixp.1
cbvixp.2
cbvixp.3

Assertion

Ref	Expression
cbvixp

Distinct variable group:   ,,

Proof of Theorem cbvixp

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvixp.1	. . . . . 6
2	1	nfel2 2637	. . . . 5
3		cbvixp.2	. . . . . 6
4	3	nfel2 2637	. . . . 5
5		fveq2 5871	. . . . . 6
6		cbvixp.3	. . . . . 6
7	5, 6	eleq12d 2539	. . . . 5
8	2, 4, 7	cbvral 3080	. . . 4
9	8	anbi2i 694	. . 3
10	9	abbii 2591	. 2
11		dfixp 7491	. 2
12		dfixp 7491	. 2
13	10, 11, 12	3eqtr4i 2496	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  F/_wnfc 2605  A.wral 2807  Fnwfn 5588  `cfv 5593  X_cixp 7489

This theorem is referenced by:  cbvixpv  7507  mptelixpg  7526  ixpiunwdom  8038  prdsbas3  14878  elptr2  20075  ptunimpt  20096  ptcldmpt  20115  finixpnum  30038  ptrest  30048

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-ral 2812  df-rex 2813  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-uni 4250  df-br 4453  df-iota 5556  df-fn 5596  df-fv 5601  df-ixp 7490