Theorem iineq1 3984

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Assertion

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

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem iineq1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 2808	. . 3 |- (A = B -> (A.x e. A y e. C <-> A.x e. B y e. C))
2	1	abbidv 2468	. 2 |- (A = B -> {y | A.x e. A y e. C} = {y | A.x e. B y e. C})
3		df-iin 3973	. 2 |- |^|_x e. A C = {y | A.x e. A y e. C}
4		df-iin 3973	. 2 |- |^|_x e. B C = {y | A.x e. B y e. C}
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> |^|_x e. A C = |^|_x e. B C)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  |^|_ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-iin 3973

This theorem is referenced by:  iinrab2  4030  riin0  4040