Theorem bj-rexcom4 33403

Description: Remove from rexcom4 3128 dependency on ax-ext 2440 and ax-13 1963 (and on df-or 370, df-tru 1377, df-sb 1707, df-clab 2448, df-cleq 2454, df-clel 2457, df-nfc 2612, df-v 3110). This proof uses only df-rex 2815 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-rexcom4	|- ( E. x e. A E. y ph <-> E. y E. x e. A ph )

Distinct variable groups:   x, y   y, A

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

Proof of Theorem bj-rexcom4

Step	Hyp	Ref	Expression
1		df-rex 2815	. 2 |- ( E. x e. A E. y ph <-> E. x ( x e. A /\ E. y ph ) )
2		19.42v 1944	. . . . 5 |- ( E. y ( x e. A /\ ph ) <-> ( x e. A /\ E. y ph ) )
3	2	bicomi 202	. . . 4 |- ( ( x e. A /\ E. y ph ) <-> E. y ( x e. A /\ ph ) )
4	3	exbii 1639	. . 3 |- ( E. x ( x e. A /\ E. y ph ) <-> E. x E. y ( x e. A /\ ph ) )
5		excom 1793	. . . 4 |- ( E. x E. y ( x e. A /\ ph ) <-> E. y E. x ( x e. A /\ ph ) )
6		df-rex 2815	. . . . . 6 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
7	6	bicomi 202	. . . . 5 |- ( E. x ( x e. A /\ ph ) <-> E. x e. A ph )
8	7	exbii 1639	. . . 4 |- ( E. y E. x ( x e. A /\ ph ) <-> E. y E. x e. A ph )
9	5, 8	bitri 249	. . 3 |- ( E. x E. y ( x e. A /\ ph ) <-> E. y E. x e. A ph )
10	4, 9	bitri 249	. 2 |- ( E. x ( x e. A /\ E. y ph ) <-> E. y E. x e. A ph )
11	1, 10	bitri 249	1 |- ( E. x e. A E. y ph <-> E. y E. x e. A ph )

Syntax hints:   <-> wb 184   /\ wa 369   E.wex 1591   e. wcel 1762   E.wrex 2810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595  df-rex 2815

This theorem is referenced by:  bj-rexcom4a  33404