Theorem bj-ralvw

Description: A weak version of ralv not using ax-ext (nor df-cleq , df-clel , df-v ), and only core FOL axioms. See also bj-rexvw . The analogues for reuv and rmov are not proved. (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-ralvw.1	|- ps
Assertion	bj-ralvw	|- ( A. x e. { y | ps } ph <-> A. x ph )

Proof

Step	Hyp	Ref	Expression
1		bj-ralvw.1	|- ps
2		df-ral	|- ( A. x e. { y | ps } ph <-> A. x ( x e. { y | ps } -> ph ) )
3	1	vexw	|- x e. { y | ps }
4	3	a1bi	|- ( ph <-> ( x e. { y | ps } -> ph ) )
5	4	albii	|- ( A. x ph <-> A. x ( x e. { y | ps } -> ph ) )
6	2 5	bitr4i	|- ( A. x e. { y | ps } ph <-> A. x ph )