Theorem iineq1 2576

Description: Equality theorem for restricted existential quantifier.

Assertion

Ref	Expression
iineq1	|- (A = B -> |^|_x e. A C = |^|_x e. B C)

Distinct variable groups:   x,A   x,B

Proof of Theorem iineq1

Step	Hyp	Ref	Expression
1		raleq1 1786	. . 3 |- (A = B -> (A.x e. A y e. C <-> A.x e. B y e. C))
2	1	abbidv 1577	. 2 |- (A = B -> {y | A.x e. A y e. C} = {y | A.x e. B y e. C})
3		df-iin 2569	. 2 |- |^|_x e. A C = {y | A.x e. A y e. C}
4		df-iin 2569	. 2 |- |^|_x e. B C = {y | A.x e. B y e. C}
5	2, 3, 4	3eqtr4g 1531	1 |- (A = B -> |^|_x e. A C = |^|_x e. B C)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  |^|_ciin 2567

This theorem is referenced by:  iin0 2740

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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-iin 2569

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