Theorem bnj956 30847

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj956.1	|- ( A = B -> A. x A = B )

Assertion

Ref	Expression
bnj956	|- ( A = B -> U_ x e. A C = U_ x e. B C )

Proof of Theorem bnj956

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj956.1	. . . 4 |- ( A = B -> A. x A = B )
2		eleq2 2690	. . . . . . 7 |- ( A = B -> ( x e. A <-> x e. B ) )
3	2	anbi1d 741	. . . . . 6 |- ( A = B -> ( ( x e. A /\ y e. C ) <-> ( x e. B /\ y e. C ) ) )
4	3	alexbii 1760	. . . . 5 |- ( A. x A = B -> ( E. x ( x e. A /\ y e. C ) <-> E. x ( x e. B /\ y e. C ) ) )
5		df-rex 2918	. . . . 5 |- ( E. x e. A y e. C <-> E. x ( x e. A /\ y e. C ) )
6		df-rex 2918	. . . . 5 |- ( E. x e. B y e. C <-> E. x ( x e. B /\ y e. C ) )
7	4, 5, 6	3bitr4g 303	. . . 4 |- ( A. x A = B -> ( E. x e. A y e. C <-> E. x e. B y e. C ) )
8	1, 7	syl 17	. . 3 |- ( A = B -> ( E. x e. A y e. C <-> E. x e. B y e. C ) )
9	8	abbidv 2741	. 2 |- ( A = B -> { y | E. x e. A y e. C } = { y | E. x e. B y e. C } )
10		df-iun 4522	. 2 |- U_ x e. A C = { y | E. x e. A y e. C }
11		df-iun 4522	. 2 |- U_ x e. B C = { y | E. x e. B y e. C }
12	9, 10, 11	3eqtr4g 2681	1 |- ( A = B -> U_ x e. A C = U_ x e. B C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   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-rex 2918  df-iun 4522

This theorem is referenced by:  bnj1316  30891  bnj953  31009  bnj1000  31011  bnj966  31014