Theorem cbvexvw 1970

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Hypothesis

Ref	Expression
cbvalvw.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvexvw	|- ( E. x ph <-> E. y ps )

Distinct variable groups:   x, y   ps, x   ph, y

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvexvw

Step	Hyp	Ref	Expression
1		cbvalvw.1	. . . . 5 |- ( x = y -> ( ph <-> ps ) )
2	1	notbid 308	. . . 4 |- ( x = y -> ( -. ph <-> -. ps ) )
3	2	cbvalvw 1969	. . 3 |- ( A. x -. ph <-> A. y -. ps )
4	3	notbii 310	. 2 |- ( -. A. x -. ph <-> -. A. y -. ps )
5		df-ex 1705	. 2 |- ( E. x ph <-> -. A. x -. ph )
6		df-ex 1705	. 2 |- ( E. y ps <-> -. A. y -. ps )
7	4, 5, 6	3bitr4i 292	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   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-an 386  df-ex 1705

This theorem is referenced by:  suppimacnv  7306