Theorem rabeqf 1811

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 1568	. . 3 |- (A = B -> A.x A = B)
4		eleq2 1538	. . . 4 |- (A = B -> (x e. A <-> x e. B))
5	4	anbi1d 619	. . 3 |- (A = B -> ((x e. A /\ ph) <-> (x e. B /\ ph)))
6	3, 5	abbid 1579	. 2 |- (A = B -> {x | (x e. A /\ ph)} = {x | (x e. B /\ ph)})
7		df-rab 1655	. 2 |- {x e. A | ph} = {x | (x e. A /\ ph)}
8		df-rab 1655	. 2 |- {x e. B | ph} = {x | (x e. B /\ ph)}
9	6, 7, 8	3eqtr4g 1534	1 |- (A = B -> {x e. A | ph} = {x e. B | ph})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  {cab 1466  {crab 1651

This theorem is referenced by:  rabeq 1812  hta 4738

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655

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