Theorem iineq1 3671

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 2505	. . 3 |- ( A = B -> ( A. x e. A y e. C <-> A. x e. B y e. C ) )
2	1	abbidv 2155	. 2 |- ( A = B -> { y | A. x e. A y e. C } = { y | A. x e. B y e. C } )
3		df-iin 3660	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
4		df-iin 3660	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
5	2, 3, 4	3eqtr4g 2097	1 |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   |^|_ciin 3658

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-iin 3660

This theorem is referenced by:  riin0  3728  iin0r  3922