Theorem iuneq12daf 29373

Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)

Hypotheses

Ref	Expression
iuneq12daf.1	|- F/ x ph
iuneq12daf.2	|- F/_ x A
iuneq12daf.3	|- F/_ x B
iuneq12daf.4	|- ( ph -> A = B )
iuneq12daf.5	|- ( ( ph /\ x e. A ) -> C = D )

Assertion

Ref	Expression
iuneq12daf	|- ( ph -> U_ x e. A C = U_ x e. B D )

Proof of Theorem iuneq12daf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq12daf.1	. . . . 5 |- F/ x ph
2		iuneq12daf.5	. . . . . 6 |- ( ( ph /\ x e. A ) -> C = D )
3	2	eleq2d 2687	. . . . 5 |- ( ( ph /\ x e. A ) -> ( y e. C <-> y e. D ) )
4	1, 3	rexbida 3047	. . . 4 |- ( ph -> ( E. x e. A y e. C <-> E. x e. A y e. D ) )
5		iuneq12daf.4	. . . . 5 |- ( ph -> A = B )
6		iuneq12daf.2	. . . . . 6 |- F/_ x A
7		iuneq12daf.3	. . . . . 6 |- F/_ x B
8	6, 7	rexeqf 3135	. . . . 5 |- ( A = B -> ( E. x e. A y e. D <-> E. x e. B y e. D ) )
9	5, 8	syl 17	. . . 4 |- ( ph -> ( E. x e. A y e. D <-> E. x e. B y e. D ) )
10	4, 9	bitrd 268	. . 3 |- ( ph -> ( E. x e. A y e. C <-> E. x e. B y e. D ) )
11	10	alrimiv 1855	. 2 |- ( ph -> A. y ( E. x e. A y e. C <-> E. x e. B y e. D ) )
12		abbi 2737	. . 3 |- ( A. y ( E. x e. A y e. C <-> E. x e. B y e. D ) <-> { y | E. x e. A y e. C } = { y | E. x e. B y e. D } )
13		df-iun 4522	. . . 4 |- U_ x e. A C = { y | E. x e. A y e. C }
14		df-iun 4522	. . . 4 |- U_ x e. B D = { y | E. x e. B y e. D }
15	13, 14	eqeq12i 2636	. . 3 |- ( U_ x e. A C = U_ x e. B D <-> { y | E. x e. A y e. C } = { y | E. x e. B y e. D } )
16	12, 15	bitr4i 267	. 2 |- ( A. y ( E. x e. A y e. C <-> E. x e. B y e. D ) <-> U_ x e. A C = U_ x e. B D )
17	11, 16	sylib 208	1 |- ( ph -> U_ x e. A C = U_ x e. B D )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   {cab 2608   F/_wnfc 2751   E.wrex 2913   U_ciun 4520

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-rex 2918  df-iun 4522

This theorem is referenced by:  measvunilem0  30276