Theorem ixpeq1 7500

Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Assertion

Ref	Expression
ixpeq1

Distinct variable groups:   ,   ,

Proof of Theorem ixpeq1

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fneq2 5675	. . . 4
2		raleq 3054	. . . 4
3	1, 2	anbi12d 710	. . 3
4	3	abbidv 2593	. 2
5		dfixp 7491	. 2
6		dfixp 7491	. 2
7	4, 5, 6	3eqtr4g 2523	1

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

This theorem is referenced by:  ixpeq1d  7501  finixpnum  30038

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-ral 2812  df-fn 5596  df-ixp 7490