Theorem bj-ralcom4 32868

Description: Remove from ralcom4 3224 dependency on ax-ext 2602 and ax-13 2246 (and on df-or 385, df-an 386, df-tru 1486, df-sb 1881, df-clab 2609, df-cleq 2615, df-clel 2618, df-nfc 2753, df-v 3202). This proof uses only df-ral 2917 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ralcom4	|- ( A. x e. A A. y ph <-> A. y A. x e. A ph )

Distinct variable groups:   x, y   y, A

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

Proof of Theorem bj-ralcom4

Step	Hyp	Ref	Expression
1		19.21v 1868	. . . . 5 |- ( A. y ( x e. A -> ph ) <-> ( x e. A -> A. y ph ) )
2	1	bicomi 214	. . . 4 |- ( ( x e. A -> A. y ph ) <-> A. y ( x e. A -> ph ) )
3	2	albii 1747	. . 3 |- ( A. x ( x e. A -> A. y ph ) <-> A. x A. y ( x e. A -> ph ) )
4		alcom 2037	. . 3 |- ( A. x A. y ( x e. A -> ph ) <-> A. y A. x ( x e. A -> ph ) )
5	3, 4	bitri 264	. 2 |- ( A. x ( x e. A -> A. y ph ) <-> A. y A. x ( x e. A -> ph ) )
6		df-ral 2917	. 2 |- ( A. x e. A A. y ph <-> A. x ( x e. A -> A. y ph ) )
7		df-ral 2917	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
8	7	albii 1747	. 2 |- ( A. y A. x e. A ph <-> A. y A. x ( x e. A -> ph ) )
9	5, 6, 8	3bitr4i 292	1 |- ( A. x e. A A. y ph <-> A. y A. x e. A ph )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   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-11 2034

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-ral 2917

This theorem is referenced by: (None)