Theorem aevdemo 27317

Description: Proof illustrating the comment of aev2 1986. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
aevdemo	|- ( A. x x = y -> ( ( E. a A. b c = d \/ E. e f = g ) /\ A. h ( i = j -> k = l ) ) )

Distinct variable group:   x, y

Proof of Theorem aevdemo

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 1983	. . . 4 |- ( A. x x = y -> A. e f = g )
2	1	19.2d 1893	. . 3 |- ( A. x x = y -> E. e f = g )
3	2	olcd 408	. 2 |- ( A. x x = y -> ( E. a A. b c = d \/ E. e f = g ) )
4		aev 1983	. . 3 |- ( A. x x = y -> A. m m = n )
5		aeveq 1982	. . . . 5 |- ( A. m m = n -> k = l )
6	5	a1d 25	. . . 4 |- ( A. m m = n -> ( i = j -> k = l ) )
7	6	alrimiv 1855	. . 3 |- ( A. m m = n -> A. h ( i = j -> k = l ) )
8	4, 7	syl 17	. 2 |- ( A. x x = y -> A. h ( i = j -> k = l ) )
9	3, 8	jca 554	1 |- ( A. x x = y -> ( ( E. a A. b c = d \/ E. e f = g ) /\ A. h ( i = j -> k = l ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   E.wex 1704

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-7 1935

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705

This theorem is referenced by: (None)