Theorem bj-rexcom4a 32870

Description: Remove from rexcom4a 3226 dependency on ax-ext 2602 and ax-13 2246 (and on df-or 385, df-sb 1881, df-clab 2609, df-cleq 2615, df-clel 2618, df-nfc 2753, df-v 3202). This proof uses only df-rex 2918 on top of first-order logic. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-rexcom4a	|- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )

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

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

Proof of Theorem bj-rexcom4a

Step	Hyp	Ref	Expression
1		bj-rexcom4 32869	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. x E. y e. A ( ph /\ ps ) )
2		19.42v 1918	. . 3 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
3	2	rexbii 3041	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )
4	1, 3	bitr3i 266	1 |- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   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-rex 2918

This theorem is referenced by:  bj-rexcom4bv  32871  bj-rexcom4b  32872