Theorem iuneq2i 2580

Description: Equality inference for indexed union.

Hypothesis

Ref	Expression
iuneq2i.1	|- (x e. A -> B = C)

Assertion

Ref	Expression
iuneq2i	|- U_x e. A B = U_x e. A C

Proof of Theorem iuneq2i

Step	Hyp	Ref	Expression
1		iuneq2 2578	. 2 |- (A.x e. A B = C -> U_x e. A B = U_x e. A C)
2		iuneq2i.1	. 2 |- (x e. A -> B = C)
3	1, 2	mprg 1700	1 |- U_x e. A B = U_x e. A C

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  U_ciun 2566

This theorem is referenced by:  iunab 2597  dfimafn2 3762  funiunfv 3866  abianfplem 3961  r1lim 4653  iundom 4812  alephlim 4864  subtop 7646

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-in 2051  df-ss 2053  df-iun 2568

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