Theorem iineq1 3791

Description: Equality theorem for restricted existential quantifier. (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 2598	. . 3 |- ( A = B -> ( A. x e. A y e. C <-> A. x e. B y e. C ) )
2	1	abbidv 2230	. 2 |- ( A = B -> { y | A. x e. A y e. C } = { y | A. x e. B y e. C } )
3		df-iin 3780	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
4		df-iin 3780	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
5	2, 3, 4	3eqtr4g 2170	1 |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1312   e. wcel 1461   {cab 2099   A.wral 2388   |^|_ciin 3778

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-iin 3780

This theorem is referenced by:  riin0  3848  iin0r  4051