Theorem hbequid 34194

Description: Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 34171.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbequid	|- ( x = x -> A. y x = x )

Proof of Theorem hbequid

Step	Hyp	Ref	Expression
1		ax-c9 34175	. 2 |- ( -. A. y y = x -> ( -. A. y y = x -> ( x = x -> A. y x = x ) ) )
2		ax7 1943	. . . . 5 |- ( y = x -> ( y = x -> x = x ) )
3	2	pm2.43i 52	. . . 4 |- ( y = x -> x = x )
4	3	alimi 1739	. . 3 |- ( A. y y = x -> A. y x = x )
5	4	a1d 25	. 2 |- ( A. y y = x -> ( x = x -> A. y x = x ) )
6	1, 5, 5	pm2.61ii 177	1 |- ( x = x -> A. y x = x )

Syntax hints:   -> wi 4   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-c9 34175

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

This theorem is referenced by:  nfequid-o  34195  equidq  34209