Theorem bj-rexcom4bv 32871

Description: Version of bj-rexcom4b 32872 with a dv condition on x , V, hence removing dependency on df-sb 1881 and df-clab 2609 (so that it depends on df-clel 2618 and df-rex 2918 only on top of first-order logic). Prefer its use over bj-rexcom4b 32872 when sufficient (in particular when V is substituted for _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rexcom4bv.1	|- B e. V

Assertion

Ref	Expression
bj-rexcom4bv	|- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )

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

Allowed substitution hints:   ph( y)   A( y)   B( y)   V( y)

Proof of Theorem bj-rexcom4bv

Step	Hyp	Ref	Expression
1		bj-rexcom4a 32870	. 2 |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ( ph /\ E. x x = B ) )
2		bj-rexcom4bv.1	. . . . 5 |- B e. V
3	2	bj-issetiv 32863	. . . 4 |- E. x x = B
4	3	biantru 526	. . 3 |- ( ph <-> ( ph /\ E. x x = B ) )
5	4	rexbii 3041	. 2 |- ( E. y e. A ph <-> E. y e. A ( ph /\ E. x x = B ) )
6	1, 5	bitr4i 267	1 |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913

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-11 2034

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-clel 2618  df-rex 2918

This theorem is referenced by: (None)