Theorem wl-aleq 33322

Description: The semantics of A. x y = z. (Contributed by Wolf Lammen, 27-Apr-2018.)

Assertion

Ref	Expression
wl-aleq	|- ( A. x y = z <-> ( y = z /\ ( A. x x = y <-> A. x x = z ) ) )

Proof of Theorem wl-aleq

Step	Hyp	Ref	Expression
1		sp 2053	. . 3 |- ( A. x y = z -> y = z )
2		equequ2 1953	. . . . 5 |- ( y = z -> ( x = y <-> x = z ) )
3	2	alimi 1739	. . . 4 |- ( A. x y = z -> A. x ( x = y <-> x = z ) )
4		albi 1746	. . . 4 |- ( A. x ( x = y <-> x = z ) -> ( A. x x = y <-> A. x x = z ) )
5	3, 4	syl 17	. . 3 |- ( A. x y = z -> ( A. x x = y <-> A. x x = z ) )
6	1, 5	jca 554	. 2 |- ( A. x y = z -> ( y = z /\ ( A. x x = y <-> A. x x = z ) ) )
7		ax7 1943	. . . . . 6 |- ( x = y -> ( x = z -> y = z ) )
8	7	al2imi 1743	. . . . 5 |- ( A. x x = y -> ( A. x x = z -> A. x y = z ) )
9	8	a1dd 50	. . . 4 |- ( A. x x = y -> ( A. x x = z -> ( y = z -> A. x y = z ) ) )
10		axc9 2302	. . . 4 |- ( -. A. x x = y -> ( -. A. x x = z -> ( y = z -> A. x y = z ) ) )
11	9, 10	bija 370	. . 3 |- ( ( A. x x = y <-> A. x x = z ) -> ( y = z -> A. x y = z ) )
12	11	impcom 446	. 2 |- ( ( y = z /\ ( A. x x = y <-> A. x x = z ) ) -> A. x y = z )
13	6, 12	impbii 199	1 |- ( A. x y = z <-> ( y = z /\ ( A. x x = y <-> A. x x = z ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481

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  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  wl-nfeqfb  33323  wl-ax11-lem2  33363