Theorem wl-lem-exsb 31965

Description: This theorem provides a basic working step in proving theorems about E* or E!. (Contributed by Wolf Lammen, 3-Oct-2019.)

Assertion

Ref	Expression
wl-lem-exsb	|- ( x = y -> ( ph <-> A. x ( x = y -> ph ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem wl-lem-exsb

Step	Hyp	Ref	Expression
1		ax12v2 1952	. 2 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
2		sp 1957	. . 3 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
3	2	com12 31	. 2 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
4	1, 3	impbid 195	1 |- ( x = y -> ( ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672

This theorem is referenced by: (None)