Theorem bnj1316 30891

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

Hypotheses

Ref	Expression
bnj1316.1	|- ( y e. A -> A. x y e. A )
bnj1316.2	|- ( y e. B -> A. x y e. B )

Assertion

Ref	Expression
bnj1316	|- ( A = B -> U_ x e. A C = U_ x e. B C )

Distinct variable groups:   y, A   y, B   x, y

Allowed substitution hints:   A( x)   B( x)   C( x, y)

Proof of Theorem bnj1316

Step	Hyp	Ref	Expression
1		bnj1316.1	. . . . 5 |- ( y e. A -> A. x y e. A )
2	1	nfcii 2755	. . . 4 |- F/_ x A
3		bnj1316.2	. . . . 5 |- ( y e. B -> A. x y e. B )
4	3	nfcii 2755	. . . 4 |- F/_ x B
5	2, 4	nfeq 2776	. . 3 |- F/ x A = B
6	5	nf5ri 2065	. 2 |- ( A = B -> A. x A = B )
7	6	bnj956 30847	1 |- ( A = B -> U_ x e. A C = U_ x e. B C )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   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:  bnj1000  31011  bnj1318  31093