Theorem iineq1 4311

Description: Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
iineq1	|- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem iineq1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3025	. . 3 |- ( A = B -> ( A. x e. A y e. C <-> A. x e. B y e. C ) )
2	1	abbidv 2558	. 2 |- ( A = B -> { y | A. x e. A y e. C } = { y | A. x e. B y e. C } )
3		df-iin 4299	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
4		df-iin 4299	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
5	2, 3, 4	3eqtr4g 2488	1 |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1437   e. wcel 1868   {cab 2407   A.wral 2775   |^|_ciin 4297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-iin 4299

This theorem is referenced by:  iinrab2  4359  iinvdif  4368  riin0  4370  iin0  4594  xpriindi  4986  cmpfi  20409  ptbasfi  20582  fclsval  21009  taylfval  23300  polvalN  33388