Theorem iuneq2f 33963

Description: Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Hypotheses

Ref	Expression
iuneq2f.1	|- F/_ x A
iuneq2f.2	|- F/_ x B

Assertion

Ref	Expression
iuneq2f	|- ( A = B -> U_ x e. A C = U_ x e. B C )

Proof of Theorem iuneq2f

Step	Hyp	Ref	Expression
1		iuneq2f.1	. . 3 |- F/_ x A
2		iuneq2f.2	. . 3 |- F/_ x B
3	1, 2	nfeq 2776	. 2 |- F/ x A = B
4		id 22	. 2 |- ( A = B -> A = B )
5		eqidd 2623	. 2 |- ( A = B -> C = C )
6	3, 1, 2, 4, 5	iuneq12df 4544	1 |- ( A = B -> U_ x e. A C = U_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1483   F/_wnfc 2751   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:  iuneq12f  33972