Theorem bj-ralvw 32865

Description: A weak version of ralv 3219 not using ax-ext 2602 (nor df-cleq 2615, df-clel 2618, df-v 3202), but using ax-13 2246. For the sake of illustration, the next theorem bj-rexvwv 32866, a weak version of rexv 3220, has a dv condition and avoids dependency on ax-13 2246, while the analogues for reuv 3221 and rmov 3222 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-ralvw.1	|- ps

Assertion

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

Proof of Theorem bj-ralvw

Step	Hyp	Ref	Expression
1		df-ral 2917	. 2 |- ( A. x e. { y | ps } ph <-> A. x ( x e. { y | ps } -> ph ) )
2		bj-ralvw.1	. . . . 5 |- ps
3	2	bj-vexw 32855	. . . 4 |- x e. { y | ps }
4	3	a1bi 352	. . 3 |- ( ph <-> ( x e. { y | ps } -> ph ) )
5	4	albii 1747	. 2 |- ( A. x ph <-> A. x ( x e. { y | ps } -> ph ) )
6	1, 5	bitr4i 267	1 |- ( A. x e. { y | ps } ph <-> A. x ph )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   {cab 2608   A.wral 2912

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-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-ral 2917

This theorem is referenced by: (None)