Theorem iineq1 4296

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 3023	. . 3 |- ( A = B -> ( A. x e. A y e. C <-> A. x e. B y e. C ) )
2	1	abbidv 2590	. 2 |- ( A = B -> { y | A. x e. A y e. C } = { y | A. x e. B y e. C } )
3		df-iin 4285	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
4		df-iin 4285	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
5	2, 3, 4	3eqtr4g 2520	1 |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   {cab 2439   A.wral 2799   |^|_ciin 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-iin 4285

This theorem is referenced by:  iinrab2  4344  iinvdif  4353  riin0  4355  iin0  4577  xpriindi  5087  cmpfi  19153  ptbasfi  19296  fclsval  19723  taylfval  21967  polvalN  33912