Theorem bj-rexcom4b 31481

Description: Remove from rexcom4b 3069 dependency on ax-ext 2431 and ax-13 2091 (and on df-or 372, df-cleq 2444, df-nfc 2581, df-v 3047). The hypothesis uses V instead of _V (see bj-isseti 31473 for the motivation). Use bj-rexcom4bv 31480 instead when sufficient (in particular when V is substituted for _V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rexcom4b.1	|- B e. V

Assertion

Ref	Expression
bj-rexcom4b	|- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )

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

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

Proof of Theorem bj-rexcom4b

Step	Hyp	Ref	Expression
1		bj-rexcom4a 31479	. 2 |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ( ph /\ E. x x = B ) )
2		bj-rexcom4b.1	. . . . 5 |- B e. V
3	2	bj-isseti 31473	. . . 4 |- E. x x = B
4	3	biantru 508	. . 3 |- ( ph <-> ( ph /\ E. x x = B ) )
5	4	rexbii 2889	. 2 |- ( E. y e. A ph <-> E. y e. A ( ph /\ E. x x = B ) )
6	1, 5	bitr4i 256	1 |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )

Syntax hints:   <-> wb 188   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   E.wrex 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-11 1920  ax-12 1933

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-sb 1798  df-clab 2438  df-clel 2447  df-rex 2743

This theorem is referenced by: (None)