Theorem equsv 1934

Description: If a formula does not contain a variable x, then it is equivalent to the corresponding prototype of substitution with a fresh variable (see sb6 1937). (Contributed by BJ, 23-Jul-2023.)

Assertion

Ref	Expression
equsv	|- ( A. x ( x = y -> ph ) <-> ph )

Distinct variable groups:   x, y   ph, x

Allowed substitution hint:   ph( y)

Proof of Theorem equsv

Step	Hyp	Ref	Expression
1		19.23v 1932	. 2 |- ( A. x ( x = y -> ph ) <-> ( E. x x = y -> ph ) )
2		a9ev 1745	. . 3 |- E. x x = y
3	2	a1bi 243	. 2 |- ( ph <-> ( E. x x = y -> ph ) )
4	1, 3	bitr4i 187	1 |- ( A. x ( x = y -> ph ) <-> ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1498  ax-ie2 1543  ax-17 1575  ax-i9 1579

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)