Theorem rabeqf 1799

Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions.

Hypotheses

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

Assertion

Ref	Expression
rabeqf	|- (A = B -> {x e. A | ph} = {x e. B | ph})

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

Proof of Theorem rabeqf

Step	Hyp	Ref	Expression
1		rabeqf.1	. . . 4 |- (y e. A -> A.x y e. A)
2		rabeqf.2	. . . 4 |- (y e. B -> A.x y e. B)
3	1, 2	hbeq 1557	. . 3 |- (A = B -> A.x A = B)
4		eleq2 1527	. . . 4 |- (A = B -> (x e. A <-> x e. B))
5	4	anbi1d 615	. . 3 |- (A = B -> ((x e. A /\ ph) <-> (x e. B /\ ph)))
6	3, 5	abbid 1568	. 2 |- (A = B -> {x | (x e. A /\ ph)} = {x | (x e. B /\ ph)})
7		df-rab 1644	. 2 |- {x e. A | ph} = {x | (x e. A /\ ph)}
8		df-rab 1644	. 2 |- {x e. B | ph} = {x | (x e. B /\ ph)}
9	6, 7, 8	3eqtr4g 1523	1 |- (A = B -> {x e. A | ph} = {x e. B | ph})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  {cab 1456  {crab 1640

This theorem is referenced by:  rabeq 1800  hta 4700

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rab 1644

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