Theorem iineq1 4307

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 2999	. . 3 |- ( A = B -> ( A. x e. A y e. C <-> A. x e. B y e. C ) )
2	1	abbidv 2580	. 2 |- ( A = B -> { y | A. x e. A y e. C } = { y | A. x e. B y e. C } )
3		df-iin 4295	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
4		df-iin 4295	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
5	2, 3, 4	3eqtr4g 2521	1 |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1455   e. wcel 1898   {cab 2448   A.wral 2749   |^|_ciin 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-iin 4295

This theorem is referenced by:  iinrab2  4355  iinvdif  4364  riin0  4366  iin0  4591  xpriindi  4990  cmpfi  20472  ptbasfi  20645  fclsval  21072  taylfval  23363  polvalN  33515