Theorem bj-rababwv 32867

Description: A weak version of rabab 3223 not using df-clel 2618 nor df-v 3202 (but requiring ax-ext 2602). A version without dv condition is provable by replacing bj-vexwv 32857 with bj-vexw 32855 in the proof, hence requiring ax-13 2246. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rababwv.1	|- ps

Assertion

Ref	Expression
bj-rababwv	|- { x e. { y | ps } | ph } = { x | ph }

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem bj-rababwv

Step	Hyp	Ref	Expression
1		df-rab 2921	. 2 |- { x e. { y | ps } | ph } = { x | ( x e. { y | ps } /\ ph ) }
2		bj-rababwv.1	. . . . 5 |- ps
3	2	bj-vexwv 32857	. . . 4 |- x e. { y | ps }
4	3	biantrur 527	. . 3 |- ( ph <-> ( x e. { y | ps } /\ ph ) )
5	4	bj-abbii 32777	. 2 |- { x | ph } = { x | ( x e. { y | ps } /\ ph ) }
6	1, 5	eqtr4i 2647	1 |- { x e. { y | ps } | ph } = { x | ph }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-rab 2921

This theorem is referenced by: (None)