Theorem iuneq12df 4544

Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)

Hypotheses

Ref	Expression
iuneq12df.1	|- F/ x ph
iuneq12df.2	|- F/_ x A
iuneq12df.3	|- F/_ x B
iuneq12df.4	|- ( ph -> A = B )
iuneq12df.5	|- ( ph -> C = D )

Assertion

Ref	Expression
iuneq12df	|- ( ph -> U_ x e. A C = U_ x e. B D )

Proof of Theorem iuneq12df

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq12df.1	. . . 4 |- F/ x ph
2		iuneq12df.2	. . . 4 |- F/_ x A
3		iuneq12df.3	. . . 4 |- F/_ x B
4		iuneq12df.4	. . . 4 |- ( ph -> A = B )
5		iuneq12df.5	. . . . 5 |- ( ph -> C = D )
6	5	eleq2d 2687	. . . 4 |- ( ph -> ( y e. C <-> y e. D ) )
7	1, 2, 3, 4, 6	rexeqbid 3151	. . 3 |- ( ph -> ( E. x e. A y e. C <-> E. x e. B y e. D ) )
8	7	alrimiv 1855	. 2 |- ( ph -> A. y ( E. x e. A y e. C <-> E. x e. B y e. D ) )
9		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 } )
10		df-iun 4522	. . . 4 |- U_ x e. A C = { y | E. x e. A y e. C }
11		df-iun 4522	. . . 4 |- U_ x e. B D = { y | E. x e. B y e. D }
12	10, 11	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 } )
13	9, 12	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 )
14	8, 13	sylib 208	1 |- ( ph -> U_ x e. A C = U_ x e. B D )

Syntax hints:   -> wi 4   <-> wb 196   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:  iunxdif3  4606  iundisjf  29402  aciunf1  29463  measvuni  30277  iuneq2f  33963