Theorem iineq12f 33973

Description: Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Hypotheses

Ref	Expression
iineq12f.1	|- F/_ x A
iineq12f.2	|- F/_ x B

Assertion

Ref	Expression
iineq12f	|- ( ( A = B /\ A. x e. A C = D ) -> |^|_ x e. A C = |^|_ x e. B D )

Proof of Theorem iineq12f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2690	. . . . . 6 |- ( C = D -> ( y e. C <-> y e. D ) )
2	1	ralimi 2952	. . . . 5 |- ( A. x e. A C = D -> A. x e. A ( y e. C <-> y e. D ) )
3		ralbi 3068	. . . . 5 |- ( A. x e. A ( y e. C <-> y e. D ) -> ( A. x e. A y e. C <-> A. x e. A y e. D ) )
4	2, 3	syl 17	. . . 4 |- ( A. x e. A C = D -> ( A. x e. A y e. C <-> A. x e. A y e. D ) )
5		iineq12f.1	. . . . 5 |- F/_ x A
6		iineq12f.2	. . . . 5 |- F/_ x B
7	5, 6	raleqf 3134	. . . 4 |- ( A = B -> ( A. x e. A y e. D <-> A. x e. B y e. D ) )
8	4, 7	sylan9bbr 737	. . 3 |- ( ( A = B /\ A. x e. A C = D ) -> ( A. x e. A y e. C <-> A. x e. B y e. D ) )
9	8	abbidv 2741	. 2 |- ( ( A = B /\ A. x e. A C = D ) -> { y | A. x e. A y e. C } = { y | A. x e. B y e. D } )
10		df-iin 4523	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
11		df-iin 4523	. 2 |- |^|_ x e. B D = { y | A. x e. B y e. D }
12	9, 10, 11	3eqtr4g 2681	1 |- ( ( A = B /\ A. x e. A C = D ) -> |^|_ x e. A C = |^|_ x e. B D )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-iin 4523

This theorem is referenced by: (None)